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ガロア理論入門

 ガロア理論入門

Etsuji Nakai

April 12, 2023
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  1. ΨϩΞཧ࿦ೖ໳
    தҪ ӻ࢘
    2023 ೥ 4 ݄ 12 ೔
    1 ͸͡Ίʹ
    ຊॻΛखʹऔͬͨօ͞ΜͰ͋Ε͹ɺؒҧ͍ͳ͘ɺΨϩΞͷੜ֔ʹ͍ͭͯࣖʹͨ͜͠ͱ͕͋ΔͰ͠ΐ͏ɻे୅
    ʹͯ͠਺ֶ࢙ʹ࢒ΔઌݧతͳݚڀΛߦ͍ͳ͕Β΋ɺ
    ʮ๻ʹ͸͕࣌ؒͳ͍ʯͱ͍͏༗໊ͳҰઅΛهͨ͠खࢴΛ࢒
    ͠ɺೋेࡀͷए͞ͰܾಆʹΑΓࢮΛܴ͑ͨ —— ͦͷ༗໊ͳҳ࿩Λ஌ͬͨ೔ɺ1 ਓࣗࣨʹ͜΋ͬͯ୅਺ֶͷڭ
    ՊॻΛ։͖ͳ͕Βɺ
    ʮΨϩΞ͕ࢮΛܴ͑ͨ೥ྸΛ௒͑ͨʹ΋ؔΘΒͣɺԶʗࢲ͸ະͩʹΨϩΞʹඖఢ͢Δݚڀ
    ੒ՌΛԿ΋͍͋͛ͯͳ͍ɾɾɾͳΜͯ͜ͱͩʂʯͱۤ೰ͨ͠໷Λࢥ͍ग़͢ಡऀ΋গͳ͘ͳ͍͸ͣͰ͢ɻ
    ͔͠͠ͳ͕ΒɺͦͷҰํͰɺΨϩΞ͕ߏஙͨ͠ཧ࿦ɺಛʹɺ
    ʮ5 ࣍Ҏ্ͷํఔࣜʹ͸Ұൠղ͕ଘࡏ͠ͳ͍ʯ͜
    ͱΛࣔ͢ͱ͍͏༗໊ͳఆཧͷ಺༰Λཧղ͍ͯ͠Δํ͸ɺҙ֎ͱଟ͘͸ͳ͍ͷ͔΋஌Ε·ͤΜɻ࣮͸ɺචऀࣗ਎
    ΋େֶੜ࣌୅ʹʮΨϩΞཧ࿦ʯͱ୊͞Εͨ਺࡭ͷॻ੶ΛಡΜͩ͜ͱ͸͋ΔͷͰ͕͢ɺ݁ہͷॴɺͦͷཧ࿦ͷ಺
    ༰ΛʮཧղͰ͖ͨʂʯͱ࣮ײͨ͠هԱ͕͋Γ·ͤΜɻͦ͜ͰɺຊߘͰ͸ɺΦϯϥΠϯͰࢀরͰ͖Δ͍͔ͭ͘ͷ
    ࢿྉΛ΋ͱʹͯ͠ɺલड़ͷఆཧͷূ໌ʹࢸΔಓےΛࣗ෼ͳΓʹ࠶ߏ੒ͯ͠Έ·ͨ͠ɻओʹࢀߟʹ͍ͤͯͨͩ͞
    ͍ͨࢿྉ͸ɺ[1][2][3] ͷ 3 ͭʹͳΓ·͢ɻ·ͨɺલఏͱ͢Δ஌ࣝ͸͜ͷ͋ͨΓͰ͢ɻ
    • ܈࿦શൠʢਖ਼ن෦෼܈ɺ४ಉܕఆཧɺಉܕఆཧɺू߹ʹର͢Δ܈ͷ࡞༻ͳͲʣ
    • ؀ͱମͷجຊʢ؀ʗମͷఆٛɺ४ಉܕࣸ૾ͷੑ࣭ͳͲʣ
    • ଟ߲ࣜͷجຊʢϢʔΫϦουͷޓআ๏ɺ৒༨ఆཧͳͲʣ
    ͦͯ͠ɺຊߘͷ໨ඪ͸ɺ࣍ͷͱ͓ΓʹͳΓ·͢ɻ
    • ଟ߲ࣜͷՄղੑͱՄղ܈ͷؔ܎Λࣔ͢༗໊ͳఆཧΛূ໌͢Δɻ
    • ূ໌ͷத਎Λཧղͯ͠ɺଟ߲ࣜͷՄղੑͱʮղͷެࣜʯͷؔ܎Λཧղ͢Δɻ
    • ͦͷ্Ͱɺ࣮ࡍʹ 3 ࣍ํఔࣜͷղͷެࣜΛߏ੒ͯ͠ΈΔɻ
    5 ࣍Ҏ্ͷํఔࣜʹ͸Ұൠղ͕ଘࡏ͠ͳ͍ɺͱ͍͏͜ͱ͸༗໊Ͱ͕͢ɺٯʹ 4 ࣍ҎԼͷํఔࣜʹ͍ͭͯ͸Ͳ
    ͏͔ͱ͍͏ͱɺલड़ͷఆཧͷओுΛද໘తʹݟ͍ͯΔ͚ͩͰ͸ɺ
    ʮҰൠղ͕ଘࡏ͠ͳ͍͜ͱ͸ͳ͍ʯͱ͍͏ͩ
    ͚Ͱɺ۩ମతʹҰൠղʢղͷެࣜʣΛߏ੒͢ΔͨΊͷΞϧΰϦζϜ͸ݟ͖͑ͯ·ͤΜɻຊߘͰ͸ɺ
    ʮఆཧͷূ
    ໌ͦͷ΋ͷΛཧղ͢Δ͜ͱͰɺҰൠղΛߏ੒͢ΔΞϧΰϦζϜ΋෼͔ΔͷͰ͸ͳ͍ͩΖ͏͔ʯͱ͍͏ظ଴ͷ΋
    ͱʹษڧͨ݁͠ՌΛ·ͱΊͯ͋Γ·͢ɻݸʑͷิ୊΍ఆཧͷূ໌ʹ͍ͭͯ͸ɺલड़ͷࢿྉʹهࡌͷূ໌Λ΄΅
    ͦͷ··ͳͧͬͨ΋ͷͱɺಠࣗʹ࠶ߏ੒ͨ͠΋ͷ͕ࠞࡏ͍ͯ͠·͢ɻͨͩ͠ɺຊߘͷ಺༰ʹෆඋ͕͋ͬͨͱ͢
    1

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  2. Ε͹ɺͻͱ͑ʹචऀͷཧղෆ଍ʹΑΔ΋ͷͰ͋Δ͜ͱΛਃ͠ఴ͓͖͑ͯ·͢*1ɻ
    2 ମͷ֦େ
    2.1 ֦େମͷ࣍਺
    Ұൠʹɺମ E ͱମ F ͕ू߹ͱͯ͠ͷแؚؔ܎ E ⊃ F Λຬ͓ͨͯ͠Γɺମͱͯ͠ͷ E ͷԋࢉΛ F ʹ੍ݶ
    ʹͨ͠΋ͷ͕ɺମ F ͷԋࢉʹҰக͢Δ࣌ɺE ͸ F ͷ֦େମͰ͋Δͱݴ͍ɺ͜ͷؔ܎Λ
    E/F
    ͱ͍͏ه߸Ͱද͠·͢ɻ͜Ε͸ɺମ F ͷߏ଄͕ମ E ʹຒΊࠐ·Ε͍ͯΔɺ΋͘͠͸ɺମ F ͕ମ E ͷ෦෼ମ
    ʹͳ͍ͬͯΔͱཧղ͢Δ͜ͱ΋Ͱ͖ΔͰ͠ΐ͏ɻ
    ͦͯ͠ɺ͜ͷ࣌ɺE ͸ F Λ܎਺ମͱ͢ΔϕΫτϧۭؒΛߏ੒͢Δ͜ͱ͕෼͔Γ·͢ɻͨͱ͑͹ɺE ͷ೚ҙ
    ͷݩ α1
    , α2
    ͱ F ͷ೚ҙͷݩ a1
    , a2
    ʹ͍ͭͯɺ໌Β͔ʹ a1
    α1
    + a2
    α2
    ∈ E ͕੒Γཱͪ·͢ɻͦͷଞͷϕΫ
    τϧۭؒͷެཧΛຬͨ͢͜ͱ΋ɺ௚઀ܭࢉͰ֬ೝ͢Δ͜ͱ͕Ͱ͖·͢ɻैͬͯɺϕΫτϧۭؒͷੑ࣭ͱͯ͠ɺ
    F ্ͷϕΫτϧۭؒ E ͷ࣍ݩ͕Ұҙʹఆ·Γ·͢ɻ͜ΕΛ֦େͷ࣍਺ͱΑͼɺ
    [E : F]
    ͱ͍͏ه߸Ͱද͠·͢ɻ
    ྫ 2-1

    Q Λ༗ཧ਺ମͱͯ͠ɺू߹ Q(

    2) Λ࣍ࣜͰఆٛ͠·͢ɻ
    Q(

    2) = {a + b

    2 | a, b ∈ Q}
    ͜ͷू߹ʹɺ࣮਺ͷ෦෼ू߹ͱͯ͠ͷࣗવͳԋࢉΛಋೖ͢ΔͱɺQ(

    2) ͸ Q ͷ֦େମʹͳΓ·͢ɻఆ
    ͔ٛΒ΋໌Β͔ͳΑ͏ʹɺ1 ͱ

    2 ͕جఈϕΫτϧΛ༩͑ΔͷͰɺ֦େͷ࣍਺͸ɺ
    [Q(

    2) : Q] = 2
    ͱͳΓ·͢ɻ

    ্هͷྫ͸ɺ༗ཧ਺ମ Q ʹରͯ͠ɺQ ্ͷ୅਺ํఔࣜ x2 − 2 = 0 ͷղΛ෇͚Ճ֦͑ͯେͨ͠΋ͷͱߟ͑
    Δ͜ͱ͕Ͱ͖·͢ɻ͜ͷΑ͏ʹɺ୅਺ํఔࣜͷղΛ෇͚Ճ͑ͳ͕ΒମΛ֦େ͍ͯ͘͠ͱ͍͏ૢ࡞͕ɺΨϩΞཧ
    ࿦ͷ̍ͭͷϙΠϯτͱͳΓ·͢ɻຊߘͰ͸ɺ͜ͷྫͱಉ༷ʹɺ༗ཧ਺ମ Q ͷ֦େମͰɺ֦େͷ࣍਺͕༗ݶͷ
    ΋ͷΛٞ࿦ͷର৅ͱ͍͖ͯ͠·͢ɻ
    ΨϩΞཧ࿦ͦͷ΋ͷ͸ɺҰൠͷମ K Λجૅͱͯ͠ߏங͢Δ͜ͱ΋Ͱ͖·͕͢ɺͦͷ৔߹͸ɺମͷඪ਺ʹ஫
    ҙ͢Δඞཁ͕͋Γ·͢ɻ۩ମతʹઆ໌͢Δͱɺn Λࣗવ਺ͱͯ͠ɺ୅਺ํఔࣜ xn − 1 = 0 ͷղΛ 1 ͷݪ࢝ n
    ৐ࠜͱ͍͍·͢ɻ༗ཧ਺ମ্ͷ୅਺ํఔࣜͰ͋Ε͹ɺ୅਺ֶͷجຊఆཧʹΑΓɺෳૉ਺ମ C ͷதʹ n ݸͷ૬
    ҧͳΔղ͕ଘࡏͯ͠ɺ͜ΕΒͷղΛ෇͚Ճ֦͑ͨେମΛߏ੒͢Δ͜ͱ͕ՄೳʹͳΓ·͢ɻҰํɺҰൠͷମΛ
    *1 Ұ౓͜ΕΛݴͬͯΈ͔ͨͬͨʂ
    2

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  3. ܎਺ͱ͢Δํఔࣜͱͨ͠৔߹ɺମͷඪ਺ʹΑͬͯɺଘࡏ͢Δղͷݸ਺͕ҟͳΓ·͢*2ɻͦͷͨΊɺͨͱ͑͹ɺ
    ʮ7.1 ΂͖֦ࠜେͱՄղ܈ʯͷิ୊ 9 Ͱ͸ɺ
    ʮ1 ͷݪ࢝ n ৐͕ࠜ n ݸଘࡏ͢Δʯͱ͍͏ࣄ࣮Λ༻͍·͕͢ɺ͜ͷ
    ෦෼ʹ͍ͭͯ͸ɺҰൠͷମʹ͸౰ͯ͸·Βͳ͍ࣄʹͳΓ·͢ɻ
    2.2 ୅਺֦େ
    F ͷ֦େମ E ͷݩ α ͕ F ্ͷଟ߲ࣜ f(x) ͷղͰ͋Δ࣌ɺ͢ͳΘͪɺf(α) = 0 ͱͳΔ F ্ͷଟ߲ࣜ
    f(x) ͕ଘࡏ͢Δ࣌ɺα ͸ F ্Ͱ୅਺తͰ͋Δͱݴ͍·͢*3ɻ͞ΒʹɺE ͷ͢΂ͯͷݩ͕ F ্Ͱ୅਺తʹ
    ͳ͍ͬͯΔ࣌ɺE ͸ F ͷ୅਺֦େͰ͋Δͱݴ͍·͢ɻͦͯ͠ɺ֦େ E/F ͷ࣍਺͕༗ݶͰ͋Ε͹ɺ͜Ε͸ɺ
    ඞͣ୅਺֦େͰ͋Δ͜ͱ͕ূ໌͞Ε·͢ɻ
    ఆཧ 1 [E : F] < ∞ ͷ࣌ɺE ͷ͢΂ͯͷݩ͸ F ্Ͱ୅਺తͰ͋Δɻ
    ʢূ໌ʣ
    [E : F] = n ͱ͢Δͱɺ೚ҙͷ α ∈ E ʹରͯ͠ɺn + 1 ݸͷݩ {1, α, α2, · · · , αn} ͸ϕΫτϧͱͯ͠Ұ࣍ै
    ଐʹͳΔɻैͬͯɺan
    αn + an−1
    αn−1 + · · · + a0
    = 0 ͱຬͨ͢ F ্ͷ܎਺ {a0
    , a1
    , · · · , an
    } ͕ଘࡏ͢Δɻ͜
    Ε͸ɺ
    f(x) = an
    xn + an−1
    xn−1 + · · · + a0
    ͱͯ͠ɺf(α) = 0 ͱͳΔࣄΛҙຯ͢ΔͷͰɺα ͸ F ্Ͱ୅਺తͰ͋Δɻ ˙
    ҰൠʹɺF ͷ֦େମ E ͷதʹ୅਺తͳݩ α ͕ଘࡏͨ͠৔߹ɺf(α) = 0 Λຬͨ͢ F ্ͷଟ߲ࣜ f(x) ͸ෳ
    ਺ଘࡏ͠·͢ɻͦͷதͰ΋ɺ࠷খ࣍਺ͷط໿ଟ߲ࣜΛ࠷খଟ߲ࣜͱݺͼɺ
    Irr(α, F)
    ͱද͠·͢ɻ࠷খଟ߲ࣜʹ͓͚Δɺ࠷େ࣍਺߲ͷ܎਺͸ 1 ʹऔΔ΋ͷͱ͠·͢ɻ·ͨɺ୅਺తͳݩ α ʹର͢
    Δ࠷খଟ߲ࣜ͸ɺҰҙʹܾ·Γ·͢ɻ
    ܥ 1 ༗ݶ࣍ݩ֦େ E/F ʹ͓͍ͯɺ೚ҙͷݩ α ∈ E ʹରͯ͠ɺ࠷খଟ߲ࣜ Irr(α, F) ͕Ұҙʹଘࡏ͢Δɻ
    ʢূ໌ʣ
    ࠷খଟ߲ࣜͷଘࡏ͸ఆཧ 1 ΑΓ໌Β͔ͳͷͰɺҰҙੑΛূ໌͢Δɻ2 ͭͷଟ߲ࣜ f(x) ͱ g(x) ͕ڞʹ࠷খ
    ଟ߲ࣜͷ৚݅Λຬͨ͢ͱͯ͠ɺr(x) = f(x) − g(x) ͱ͢Δͱɺf(x) ͱ g(x) ͸࠷େ࣍਺߲ͷ܎਺͕ڞʹ 1 ͳ
    ͷͰɺr(x) ͷ࣍਺͸ɺf(x), g(x) ΑΓ΋௿͘ͳΔɻҰํɺఆٛΑΓɺr(α) = 0 Λຬͨ͢ͷͰɺr(x) ͕߃౳త
    ʹ 0 Ͱͳ͚Ε͹ɺf(x), g(x) ͕࠷খଟ߲ࣜͰ͋Δͱ͍͏લఏʹໃ६͢Δɻैͬͯɺr(x) ͸߃౳తʹ 0 Ͱ͋Γɺ
    f(x) ͱ g(x) ͸Ұக͢Δɻ ˙
    *2 ମ K ʹ͓͍ͯ n × 1 = 0 ͱͳΔࣗવ਺ n ͕ଘࡏ͢Δ࣌ɺ͜ͷΑ͏ͳࣗવ਺ͷ࠷খ஋Λମ K ͷඪ਺ͱݴ͍·͢ɻ͜ͷΑ͏ͳࣗ
    વ਺͕ଘࡏ͠ͳ͍৔߹͸ɺඪ਺͸ 0 Ͱ͋Δͱఆٛ͠·͢ɻඪ਺ p ̸= 0 ͷମʹ͓͍ͯɺn ͕ p ͰׂΓ੾ΕΔ৔߹ɺn = prm ͱ͓
    ͍ͯɺxn − 1 = (xm)pr
    − 1 = (xm − 1)pr ͱ͍͏Ҽ਺෼ղ͕੒Γཱͪ·͢ɻैͬͯɺ1 ͷ n ৐ࠜ͸ɺ1 ͷ m ৐ࠜʹҰகͯ͠ɺ
    ͦͷݸ਺͸ m ݸ͔͠ଘࡏ͠ͳ͍͜ͱʹͳΓ·͢ɻ
    *3 F ্ͷଟ߲ࣜͱ͸ɺF ͷݩΛ܎਺ͱ͢Δଟ߲ࣜͷࣄͰ͢ɻ
    3

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  4. ྫ 2-2

    ྫ 2-1 ͷ Q(

    2) ʹ͓͍ͯɺ

    2 ͷ࠷খଟ߲ࣜ͸ɺ
    f(x) = x2 − 2 (1)
    Ͱ༩͑ΒΕ·͢ɻ

    ઌ΄Ͳͷྫ 2-1 ʹ͓͍ͯɺ֦େͷ࣍਺ [Q(

    2) : Q] ͕ 2 Ͱ͋Δࣄ͸ɺಛʹݫີʹ͸ূ໌͍ͯ͠·ͤΜͰ͠
    ͕ͨɺ΋͠ূ໌͢ΔͷͰ͋Ε͹ɺ࣍ͷྲྀΕʹͳΔͰ͠ΐ͏ɻ·ͣɺ༗ཧ਺ͱ

    2 Λ૊Έ߹Θͤͨ਺Λ༻͍ͯɺ
    ೚ҙͷ࢛ଇԋࢉΛߦͬͨ৔߹ɺܭࢉ݁Ռ͸͔ͳΒͣɺ
    a + b

    2 (a, b ∈ Q) (2)
    ͱ͍͏ܗʹ·ͱ·Γ·͢ɻ͜ΕʹΑΓɺ͔֬ʹ 1 ͱ

    2 ͕جఈϕΫτϧʹͳ͓ͬͯΓɺ֦େͷ࣍਺͸ 2 ʹͳ
    Δࣄ͕ࣔ͞Ε·͢ɻ
    ͦΕͰ͸ɺͳͥɺ࢛ଇԋࢉͷ݁Ռ͸ඞͣ (2) ͷܗʹͳΔͷͰ͠ΐ͏͔ʁ —— ͦͷཧ༝͸ɺ࠷খଟ߲ࣜʹ͋
    Γ·͢ɻ(1) ʹରͯ͠ f(

    2) = 0 ͕੒Γཱͭ͜ͱ͔Βɺ(

    2)2 = 2 ͱ͍͏ஔ͖׵͕͑Ͱ͖ΔͷͰɺܭࢉͷ్
    தͰ

    2 ͷߴ࣍ͷ߲͕ग़͖ͯͨͱͯ͠΋ɺ࠷ऴతʹ͸ɺ

    2 ͷ 1 ࣍ͷ߲͔͠࢒Βͳ͘ͳΔͱ͍͏Θ͚Ͱ͢ɻ·
    ͨɺ೚ҙͷݩ a + b

    2 ʹରͯ͠ɺੵͷٯݩ͕ଘࡏ͢Δ͜ͱ΋ɺf(

    2) = 0 ͱ͍͏৚͔݅Βࣔ͢͜ͱ͕Ͱ͖·
    ͢*4ɻ͜ͷ݁ՌΛҰൠԽ͢Δͱɺମ F ʹݩ α Λ෇͚Ճ֦͑ͨେମ F(α) ʹ͓͍ͯɺ֦େͷ࣍਺ [F(α) : F]
    ͸ɺα ͷ࠷খଟ߲ࣜͷ࣍਺ n ʹΑܾͬͯ·Δ΋ͷͱ૝૾͢Δ͜ͱ͕Ͱ͖·͢ɻ͜Ε͕࣮ࡍʹਖ਼͍͠ࣄΛࣔ͢
    ͷ͕ɺ࣍ͷఆཧʹͳΓ·͢ɻ
    ఆཧ 2 F ͷ֦େମ E ʹ͓͍ͯɺα ∈ E Λ୅਺తͳݩͱͯ͠ɺͦͷ࠷খଟ߲ࣜ Irr(α, F) ͷ࣍਺Λ n ͱ͢
    Δɻ͜ͷ࣌ɺ࣍ͷ 3 ͭͷࣄ࣮͕੒ཱ͢Δɻ
    (a) ू߹ F(α) Λ࣍ࣜͰఆٛͯ͠ɺ
    F(α) = {a0
    + a1
    α + a2
    α2 + · · · + an−1
    αn−1 | a0
    , · · · , an−1
    ∈ F}
    ମ E ͷ෦෼ू߹ͱͯ͠ͷࣗવͳԋࢉΛಋೖ͢Δͱɺ͜Ε͸ F ͷ֦େମͱͳΔɻ͜͜ͰɺF(α) ͷݩΛ༻͍ͨ
    ԋࢉΛߦ͏ࡍʹɺα ͷ n ࣍Ҏ্ͷ߲͕ग़ݱͨ͠৔߹͸ɺp(x) Λ࠷খଟ߲ࣜ Irr(α, F) ͱͯ͠ɺp(α) = 0 ͷ৚
    ͔݅Β n − 1 ࣍ҎԼͷ߲ʹॻ͖௚͢΋ͷͱ໿ଋ͢Δɻ
    (b) ମͷ֦େ F(α)/F ʹ͓͚Δ֦େͷ࣍਺͸ɺα ͷ࠷খଟ߲ࣜͷ࣍਺ n ʹҰக͢Δɻ
    [F(α) : F] = n
    (c) ମ F(α) ͸ɺମ F ͷ୅਺֦େͰ͋Δɻ
    ʢূ໌ʣ
    (a) F(α) ͕ମͷެཧΛຬͨ͢͜ͱΛ֬ೝ͢ΔͨΊʹɺ೚ҙͷ z ∈ F(α) ʹରͯ͠ɺੵͷٯݩ͕ଘࡏ͢Δ͜ͱ
    Λࣔ͢ɻ
    ʢͦͷଞͷମͷެཧ͕੒Γཱͭ͜ͱ͸ɺఆ͔ٛΒ༰қʹ֬ೝͰ͖Δɻ
    ʣม਺ x ͷ n − 1 ࣍ҎԼͷଟ߲
    *4 ূ໌͸ɺ͙͢ޙͷఆཧ 2 Λࢀরɻ
    4

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  5. ࣜશମͷू߹Λ F[x] ͱ͢Δ࣌ɺू߹ F(α) ͷఆٛΑΓɺ೚ҙͷ z ∈ F(α) ʹରͯ͠ɺz = g(α) ͱͳΔଟ߲ࣜ
    g(x) ∈ F[x] ͕ଘࡏ͢ΔɻҰํɺα ͷ࠷খଟ߲ࣜΛ f(x) ͱ͢Δͱɺg(x) ͷ࣍਺͸ f(x) ͷ࣍਺ n ΑΓখ͞
    ͘ɺ͔ͭɺf(x) ͸ط໿ଟ߲ࣜͳͷͰɺg(x) ͱ f(x) ͷ࠷େެ໿ࣜ͸ 1 ͱͳΓɺϢʔΫϦουͷޓআ๏ΑΓɺ
    ࣍Λຬͨ͢ଟ߲ࣜ a(x), b(x) ∈ F[x] ͕ଘࡏ͢Δɻ
    f(x)a(x) + g(x)b(x) = 1
    ͜Εʹ x = α Λ୅ೖ͢Δͱɺf(α) = 0 ΑΓɺ
    g(α)b(α) = 1
    ͕੒Γཱͭɻ͜Ε͸ɺb(α) ͕ z = g(α) ͷٯݩͰ͋Δ͜ͱΛ͍ࣔͯ͠Δɻ
    (b) n ݸͷݩ {1, α, α2, · · · , αn−1} ͕ F(α) ͷجఈͱͳΔ͜ͱΛࣔ͢ɻ·ͣɺF(α) ͷఆٛΑΓɺ೚ҙͷݩ͕
    ͜ΕΒͷҰ࣍݁߹Ͱॻ͚Δ͜ͱ͸ࣗ໌ɻ࣍ʹɺ͜ΕΒ͕Ұ࣍ಠཱͰ͸ͳ͍ͱԾఆ͢Δͱɺ
    a0
    + a1
    α + a2
    α2 + · · · + an−1
    αn−1 = 0
    ͱͳΔ܎਺ a0
    , a1
    , · · · , an−1
    ∈ F ͕ଘࡏ͢Δ͜ͱʹͳΔɻ͜Ε͸ɺn − 1 ࣍ҎԼͷଟ߲ࣜ
    g(x) = a0
    + a1
    x + a2
    x2 + · · · + an−1
    xn−1
    ͕ g(α) = 0 Λຬͨ͢͜ͱΛҙຯ͓ͯ͠Γɺf(x) ͕࠷খଟ߲ࣜͰ͋Δͱ͍͏ࣄ࣮ʹໃ६͢Δɻैͬͯɺઌ΄Ͳ
    ͷ n ݸͷݩ͸Ұ࣍ಠཱͰ͋ΓɺF(α) ͷجఈͱͳΔɻ
    (c) (b) ͷ݁ՌΛఆཧ 1 ʹద༻͢Δ͜ͱͰಘΒΕΔɻ ˙
    ྫ 2-1 ʹࣔͨ͠ Q(

    2) ͕ Q ͷ֦େମʹͳΔ͜ͱ͸ɺఆཧ 2 ʹΑΓอূ͞ΕΔ͜ͱʹͳΓ·͢ɻ͢ͳΘͪɺ
    R Λ࣮਺ମͱͯ͠ɺମͷ֦େ R/Q Λߟ͑Δ࣌ɺ

    2 ∈ R ͸ɺf(x) = x2 − 2 Λ࠷খଟ߲ࣜͱ͢Δ୅਺తͳݩ
    ʹͳΓ·͢ɻैͬͯɺQ(

    2) ͸ Q ͷ֦େମͰ͋Γɺ࠷খଟ߲ࣜͷ࣍਺͕ n = 2 Ͱ͋Δ͜ͱ͔Βɺ֦େͷ࣍
    ਺͸ [Q(

    2) : Q] = 2 ʹͳΓ·͢ɻ
    ͜ͷଞʹ͸ɺ࣍ͷΑ͏ͳྫ͕ߟ͑ΒΕ·͢ɻ
    5

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  6. ྫ 2-3

    1. ମͷ֦େ R/Q ʹ͓͍ͯɺ 3

    2 ∈ R ͸ 3 ࣍ଟ߲ࣜ f(x) = x3 − 2 Λ࠷খଟ߲ࣜͱ͢Δ୅਺తͳݩ
    ʹͳ͍ͬͯΔͷͰɺ֦େମ Q( 3

    2) ʹ͓͚Δ֦େͷ࣍਺͸ɺ
    [Q( 3

    2) : Q] = 3
    ͱܾ·Γ·͢ɻΑΓ۩ମతʹ͸ɺ
    Q( 3

    2) = {a + b 3

    2 + c( 3

    2)2 | a, b, c ∈ Q}
    ͱදΘ͢͜ͱ͕Ͱ͖·͢ɻ
    2. C Λෳૉ਺ମͱ͢Δ࣌ɺମͷ֦େ C/Q ʹ͓͍ͯɺ1 ͷෳૉࡾ৐ࠜΛ ω ∈ C ͱ͢Δͱɺ͜ͷݩͷ
    ࠷খଟ߲ࣜ͸ɺ
    f(x) = x2 + x + 1
    Ͱ༩͑ΒΕ·͢ɻैͬͯɺ֦େମ Q(ω) ʹ͓͚Δ֦େͷ࣍਺͸ɺ
    [Q(ω) : Q] = 2
    ͱͳΓ·͢ɻΑΓ۩ମతʹ͸ɺ
    Q(ω) = {a + bω | a, b ∈ Q}
    ͱදΘ͢͜ͱ͕Ͱ͖·͢ɻ

    ্هͷྫΛҰൠԽ͢Δͱɺ༗ཧ਺ Q Λ܎਺ͱ͢Δ n ࣍ͷن໿ଟ߲ࣜ f(x) ʹ͓͍ͯɺf(x) = 0 ͷղΛ
    α ∈ C ͱ͢Δ࣌ɺ
    Q(α) = {a0
    + a1
    α + a2
    α2 + · · · + an−1
    αn−1 | a0
    , a1
    , · · · , an−1
    ∈ Q}
    ͸ɺ৽ͨͳ֦େମͱͳΓɺ֦େͷ࣍਺͸ɺ
    [Q(α) : Q] = n
    Ͱ༩͑ΒΕΔ͜ͱ͕෼͔Γ·͢ɻ
    ଓ͍ͯɺ༗ཧ਺ମ Q Λ 2 ஈ֊Ͱ֦େ͢ΔྫΛߟ͑ͯΈ·͢ɻ
    6

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  7. ྫ 2-4

    ͸͡Ίʹɺ༗ཧ਺ମ Q ʹ

    2 Λ෇Ճͯ͠ɺ֦େମ
    Q(

    2)/Q
    Λߏ੒͠·͢ɻ2 ࣍ͷଟ߲ࣜ f(x) = x2 − 2 Λ࠷খଟ߲ࣜͱ͢ΔݩΛ෇͚Ճ͍͑ͯΔͷͰɺ͜ͷ࣌ͷ֦େ
    ͷ࣍਺͸ɺ
    [Q(

    2) : Q] = 2
    ͱͳΓ·͢ɻଓ͍ͯɺ͜ͷ֦େମ Q(

    2) ʹରͯ͠ɺ

    3 Λ͞Βʹ෇Ճͯ͠ɺ֦େମ
    Q(

    2,

    3)/Q(

    2)
    Λߏ੒͠·͢ɻ͜ͷ֦େମͷݩ͸ɺ࣍ͷΑ͏ʹදΘ͢͜ͱ͕Ͱ͖·͢ɻ
    Q(

    2,

    3) = {a + b

    3 | a, b ∈ Q(

    2)}
    = {a + b

    2 + c

    3 + d

    6 | a, b, c, d ∈ Q} (3)
    2 ஈ֊໨ͷ֦େʹ͓͍ͯ͸ɺ2 ࣍ͷଟ߲ࣜ f(x) = x2 − 3 Λ࠷খଟ߲ࣜͱ͢ΔݩΛ෇͚Ճ͍͑ͯΔͷͰɺ
    ֦େͷ࣍਺͸ɺ
    [Q(

    2,

    3) : Q(

    2)] = 2
    ͱͳΓ·͢ɻҰํɺQ(

    2,

    3) ͸༗ཧ਺ମ Q ʹର͢Δ֦େʹ΋ͳ͓ͬͯΓɺ(3) ͷද͔ࣜΒɺ
    [Q(

    2,

    3) : Q] = 4
    Ͱ͋Δ͜ͱ͕෼͔Γ·͢ɻ͜ΕΒΛ·ͱΊΔͱɺ֦େͷ࣍਺ʹ͍ͭͯɺ࣍ͷؔ܎͕੒Γཱ͍ͬͯΔ͜ͱ͕
    ෼͔Γ·͢ɻ
    [Q(

    2,

    3) : Q] = [Q(

    2,

    3) : Q(

    2)] × [Q(

    2) : Q]

    ্هͷྫΛҰൠԽͨ͠΋ͷ͕ɺ࣍ͷఆཧʹͳΓ·͢ɻ
    ఆཧ 3 2 ஈ֊ͷମͷ֦େ M/Fɺ͓ΑͼɺE/M ͕͋ͬͨ࣌ɺͦΕͧΕͷ֦େͷ࣍਺͕༗ݶͰ͋Ε͹ɺ࣍ͷؔ
    ܎͕੒Γཱͭɻ
    [E : F] = [E : M][M : F]
    ʢূ໌ʣ
    [E : M] = m ͱͯ͠ɺE ͷ M ্ͷجఈΛ {α1
    , · · · , αm
    } ͱ͢Δɻಉ͘͡ɺ[M : F] = n ͱͯ͠ɺM ͷ F
    ্ͷجఈΛ {β1
    , · · · , βn
    } ͱ͢Δɻ͜ͷ࣌ɺ೚ҙͷ x ∈ E ʹ͍ͭͯɺ
    x =
    m
    i=1
    ai
    αi
    (ai
    ∈ M)
    ͱॻ͚ͯɺ͞ΒʹɺͦΕͧΕͷ܎਺ ai
    ∈ M ʹ͍ͭͯɺ
    ai
    =
    n
    j=1
    aij
    βj
    (aij
    ∈ F)
    7

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  8. ͱॻ͚Δɻैͬͯɺ
    x =
    m
    i=1
    n
    j=1
    aij
    αi
    βj
    (aij
    ∈ F)
    ͱͳΓɺ͜Ε͸ɺn × m ݸͷ E ͷݩ͔ΒͳΔू߹
    {αi
    βj
    | 1 ≤ i ≤ m, 1 ≤ j ≤ n} (4)
    ͕ F ্ͷϕΫτϧۭؒ E ͷશମΛுΔ͜ͱΛҙຯ͢Δɻ͜ΕΒ͕Ұ࣍ಠཱͰ͋Ε͹ɺF ্ͷϕΫτϧۭؒ
    E ͷجఈͰ͋Δ͜ͱʹͳΓɺఆཧ͕ূ໌͞ΕΔɻͦ͜Ͱɺ
    m
    i=1
    n
    j=1
    aij
    αi
    βj
    = 0 (aij
    ∈ F)
    ͱԾఆ͢Δͱɺ
    ai
    =
    n
    j=1
    aij
    βj
    (5)
    ͱஔ͍ͯɺ
    m
    i=1
    ai
    αi
    = 0
    ͕੒ΓཱͭͷͰɺ{α1
    , · · · , αm
    } ͕ E ͷ M ্ͷجఈͰ͋Δ͜ͱ͔Βɺai
    = 0 (i = 1, · · · , m) ͕ಘΒΕΔɻ
    ͜ͷ࣌ɺ(5) ΑΓɺ
    n
    j=1
    aij
    βj
    = 0
    ͕੒Γཱͪɺ{β1
    , · · · , βn
    } ͕ M ͷ F ্ͷجఈͰ͋Δ͜ͱ͔Βɺaij
    = 0 (1 ≤ i ≤ m, 1 ≤ j ≤ n) ͕ಘΒΕ
    Δɻ͜ΕͰɺ(4) ͕Ұ࣍ಠཱͰ͋Δ͜ͱ͕ࣔ͞Εͨɻ ˙
    3 ΨϩΞ܈
    3.1 ମͷࣗݾಉܕ܈ͱΨϩΞ܈
    ମ E ʹ͍ͭͯɺE ͔Β E ΁ͷ؀ͱͯ͠ͷࣗݾಉܕࣸ૾શମΛ
    Aut(E)
    ͱ͍͏ه߸Ͱද͠·͢ɻϕ ∈ Aut(E) ͸ɺα1
    , α2
    ∈ E ʹରͯ͠ɺ࣍ͷ৚݅Λຬͨ͢શ୯ࣹͷࣸ૾ʹͳΓ·͢ɻ
    ϕ(α1
    + α2
    ) = ϕ(α1
    ) + ϕ(α2
    ) (6)
    ϕ(α1
    α2
    ) = ϕ(α1
    )ϕ(α2
    ) (7)
    ϕ ∈ Aut(E) ͸શ୯ࣹͷࣸ૾Ͱ͋Δࣄ͔Βɺ
    ٯࣸ૾ ϕ−1 ∈ Aut(E) ͕Ұҙʹଘࡏ͠·͢ɻ͜ΕʹΑΓɺ
    Aut(E)
    ͸ɺࣸ૾ͷ߹੒Λੵͱͯ͠܈Λߏ੒͠·͢ɻ͜ΕΛମ E ͷࣗݾಉܕ܈ͱݺͼ·͢ɻ
    ͞Βʹɺମ F ͷ֦େମ E ʹ͓͍ͯɺF ͷݩΛಈ͔͞ͳ͍ࣗݾಉܕࣸ૾શମɺ͢ͳΘͪɺू߹
    {ϕ ∈ Aut(E) | ∀x ∈ F; ϕ(x) = x}
    8

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  9. Λ
    Aut(E/F)
    ͱ͍͏ه߸Ͱද͠·͢ɻ͜Ε͸ɺAut(E) ͷ෦෼܈ʹͳ͓ͬͯΓɺ͜ΕΛ֦େ E/F ͷΨϩΞ܈ͱݺͼ·͢ɻ
    ྫ 3-1

    ༗ཧ਺ମ Q ʹ

    2 Λ෇͚Ճ֦͑ͨେମ
    Q(

    2) = {a + b

    2 | a, b ∈ Q}
    ͷΨϩΞ܈ Aut(Q(

    2)/Q) ʹ͍ͭͯߟ͑ͯΈ·͢ɻQ ͷݩΛಈ͔͞ͳ͍ͱ͍͏৚݅Λߟྀ͢Δͱɺ୯
    Ґݩ 1ʢ߃౳ࣸ૾ʣͷଞʹ͸ɺ࣍ͷࣸ૾ ϕ ͕ΨϩΞ܈ʹؚ·ΕΔ͜ͱʹͳΓ·͢ɻ
    ϕ(a + b

    2) = a − b

    2
    ͜Ε͕ Q ͷݩΛಈ͔͞ͳ͍͜ͱͱɺ(6) Λຬͨ͢͜ͱ͸ࣗ໌Ͱ͢ɻ(7) ʹ͍ͭͯ͸ɺ࣍ͷܭࢉͰ֬ೝ͢Δ
    ͜ͱ͕Ͱ͖·͢ɻ
    ϕ (a + b

    2)(c + d

    2) = ϕ (ac + 2bd) + (ad + bc)

    2 = (ac + 2bd) − (ad + bc)

    2
    ϕ(a + b

    2)ϕ(c + d

    2) = (a − b

    2)(c − d

    2) = (ac + 2bd) − (ad + bc)

    2
    ·ͨɺ͙͢ʹ෼͔ΔΑ͏ʹɺࣸ૾ ϕ ͸ɺϕ ◦ ϕ = 1 ͱ͍͏ؔ܎Λຬͨ͠·͢ɻैͬͯɺΨϩΞ܈͸ɺ{1, ϕ}
    ͱ͍͏ 2 ͭͷཁૉ͔ΒͳΔ܈Ͱ͋Γɺͦͷߏ଄͸ 2 ࣍ͷରশ܈ S2
    ͱಉܕʹͳΓ·͢ɻΨϩΞ܈͕͜ΕΒ
    Ҏ֎ͷཁૉΛ࣋ͨͳ͍͜ͱͷݫີͳূ໌͸ɺޙ΄Ͳʮ4.1 ΨϩΞ֦େͷதؒମʯͷิ୊ 1 Ͱߦ͍·͢ɻ

    ্هͷྫͰ͸ɺΨϩΞ܈ Aut(Q(

    2)/Q) ͷཁૉ ϕ ͸ɺํఔࣜ x2 − 2 = 0 ͷ 2 ݸͷղ x = ±

    2 Λޓ͍
    ʹೖΕସ͑Δࣸ૾ʹͳ͍ͬͯΔ͜ͱ͕෼͔Γ·͢ɻ࣮͸ɺ͜Ε͸ඞવతͳ݁ՌͰ͢ɻͳͥͳΒɺα ͕ଟ߲ࣜ
    f(x) Λ༻͍ͨํఔࣜ f(x) = 0 ͷղͰ͋ΔͳΒ͹ɺΨϩΞ܈ͷཁૉ ϕ ͕ಉܕࣸ૾Ͱ͋Δ͜ͱ͔Βɺ
    f(ϕ(α)) = ϕ(f(α)) = ϕ(0) = 0
    ͕੒Γཱͪɺϕ(α) ΋ඞͣ f(x) = 0 ͷղʹͳΔ͔ΒͰ͢ɻ͜ͷߟ͑ํ͸ɺQ(

    2,

    3) ͷΑ͏ʹɺෳ਺ͷݩΛ
    ෇͚Ճ֦͑ͨେମʹ΋Ԡ༻͢Δ͜ͱ͕ՄೳͰ͢ɻ
    9

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  10. ྫ 3-2

    ༗ཧ਺ମ Q ʹ

    2 ͱ

    3 Λ෇͚Ճ֦͑ͨେମ
    Q(

    2,

    3) = {a + b

    2 + c

    3 + d

    6 | a, b, c, d ∈ Q}
    ͷΨϩΞ܈ Aut(Q(

    2,

    3)/Q) Λߟ͑·͢ɻQ(

    2,

    3) ͸ɺ༗ཧ਺ମ Q ʹํఔࣜ x2 − 2 = 0 ͷղ
    ±

    2ɺ͓Αͼɺํఔࣜ x2 − 3 = 0 ͷղ ±

    3 Λ෇͚Ճ͑ͨ΋ͷͱߟ͑Δ͜ͱ͕Ͱ͖·͢ɻͦ͜ͰɺͦΕ
    ͧΕͷղΛೖΕସ͑Δࣸ૾Λߟ͑Δͱɺશମͱͯ࣍͠ͷ 4 ͭͷ૊Έ߹Θ͕ͤ͋Γ·͢ɻ
    • 1 :

    2 →

    2,

    3 →

    3 ʢͲͪΒ΋ೖΕସ͑ͳ͍ɻ
    ʣ
    • ϕ1
    :

    2 → −

    2,

    3 →

    3 ʢ

    2 ͚ͩೖΕସ͑Δɻ
    ʣ
    • ϕ2
    :

    2 →

    2,

    3 → −

    3 ʢ

    3 ͚ͩೖΕସ͑Δɻ
    ʣ
    • ϕ3
    :

    2 → −

    2,

    3 → −

    3 ʢ྆ํͱ΋ೖΕସ͑Δɻ
    ʣ
    ͜ΕΒͷࣸ૾͕ Q ͷݩΛಈ͔͞ͳ͍͜ͱͱɺ(6) Λຬͨ͢͜ͱ͸ࣗ໌Ͱ͢ɻ(7) Λຬͨ͢͜ͱ΋ྫ 3-1
    ͱಉ༷ͷ௚઀ܭࢉͰ֬ೝ͢Δ͜ͱ͕Ͱ͖·͢ɻैͬͯɺ͜ΕΒ͸ɺQ ͷݩΛಈ͔͞ͳ͍ࣗݾಉܕࣸ૾Ͱ
    ͋ΓɺΨϩΞ܈ͷཁૉͱͳΓ·͢ɻ·ͨɺ{1, ϕ1
    , ϕ2
    , ϕ3
    } ͕ߏ੒͢Δ܈ͷߏ଄͸ɺΫϥΠϯͷ 4 ݩ܈ͱಉ
    ܕʹͳΓ·͢ɻྫ 3-1 ͱಉ༷ʹɺΨϩΞ܈͕͜ΕΒҎ֎ͷཁૉΛ࣋ͨͳ͍͜ͱͷূ໌͸ɺิ୊ 1 Ͱߦ͍
    ·͢ɻ

    ্هͷྫͰ͸ɺ
    xn − a = 0 (8)
    ͱ͍͏ܗͷํఔࣜʹ͍ͭͯɺͦͷղΛೖΕସ͑Δૢ࡞Λώϯτʹͯ͠ΨϩΞ܈ Aut(E/F) ͷཁૉΛൃݟ͠·
    ͨ͠ɻ·ͨɺͦͷΑ͏ʹͯ͠ಘΒΕͨΨϩΞ܈ʹ͸ɺྫ 3-1 ͷରশ܈ S2
    ɺ͋Δ͍͸ɺྫ 3-2 ͷΫϥΠϯͷ 4
    ݩ܈ͷΑ͏ʹൺֱతʹγϯϓϧͳߏ଄͕͋Γ·ͨ͠ɻ࣮͸ɺҰൠʹɺମͷ֦େʹ൐͏ΨϩΞ܈ͷߏ଄͔Βɺͦ
    ͷ֦େͷੑ࣭Λௐ΂Δ͜ͱ͕ՄೳʹͳΓ·͢ɻΑΓ۩ମతʹݴ͏ͱɺҰൠʹ (8) ͷղΛ΂͖ࠜͱݺͼ·͕͢ɺ
    ͋Δ֦େ͕ɺ΂͖ࠜΛ෇͚Ճ͑ͯಘΒΕ֦ͨେͰ͋Δ͔Ͳ͏͔͕൑ఆͰ͖ΔΑ͏ʹͳΓ·͢ɻͨͩ͠ɺ͜ͷ಺
    ༰Λਖ਼֬ʹཧղ͢Δʹ͸ɺΨϩΞ܈ͷੑ࣭Λ΋͏গ͠ৄ͘͠ௐ΂͍ͯ͘ඞཁ͕͋Γ·͢ɻ࣍અͰ͸ɺ·ͣ͸ɺ
    ࣗݾಉܕ܈ Aut(E) ͷ༗ݶ෦෼܈ͱΨϩΞ܈ͷؔ܎Λ੔ཧ͠·͢ɻ
    3.2 ࣗݾಉܕ෦෼܈͔Βੜ੒͞ΕΔ෦෼ମ
    ମ E ͷࣗݾಉܕ܈ Aut(E) ͷ༗ݶ෦෼܈ G ͕͋ͬͨ৔߹ɺG Ͱݻఆ͞ΕΔ෦෼ू߹
    EG = {x ∈ E | ∀ϕ ∈ G; ϕ(x) = x}
    ͸ɺE ͷ෦෼ମͱͳΓ·͢ɻ͜Ε͸ɺϕ ∈ G ͕ (6)(7) Λຬͨ͢͜ͱ͔Β༰қʹ֬ೝͰ͖·͢ɻҰํɺEG ͷ
    ݩΛݻఆ͢Δࣗݾಉܕࣸ૾͸ɺG ͷཁૉͷଞʹ΋ଘࡏ͢ΔՄೳੑ͕͋Γɺͦͷશମ͕ Aut(E/EG) Ͱ͋Δ͜
    ͱ͔Βɺ࣍ͷแؚؔ܎͕੒Γཱͪ·͢ɻ
    G ⊆ Aut(E/EG)
    ࣮͸͜ͷ࣌ɺG = Aut(E/EG) ͕੒ཱͯ͠ɺ͞Βʹ [E : EG] = |G| ͕੒Γཱͪ·͢ɻ͜ͷޙ͸ɺ͜ͷࣄ࣮Λ
    ॱΛ௥ͬͯࣔͯ͠ߦ͖·͢ɻ
    10

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  11. ͸͡Ίʹɺࣗݾಉܕࣸ૾ͷઢܗ݁߹Ͱɺ͢΂ͯͷݩΛ 0 ʹࣸ͢Α͏ͳࣸ૾͸ߏ੒Ͱ͖ͳ͍ͱ͍͏ఆཧΛࣔ͠
    ·͢ɻ
    ఆཧ 4 ʢσσΩϯτͷఆཧʣ
    ମ E ͷ૬ҧͳΔࣗݾಉܕࣸ૾ ϕ1
    , · · · , ϕn
    ∈ Aut(E) ͕༩͑ΒΕͨ࣌ɺa1
    , · · · , an
    ∈ E ʹ͍ͭͯɺ
    ∀x ∈ E;
    n
    i=1
    ai
    ϕi
    (x) = 0
    ͕੒ΓཱͭͳΒ͹ɺ
    ai
    = 0 (i = 1, · · · , n)
    Ͱ͋Δɻ
    ʢূ໌ʣ
    n ʹ͍ͭͯͷؼೲ๏Ͱࣔ͢ɻn = 1 ͷ࣌ɺϕ1
    ͸ࣗݾಉܕࣸ૾ͳͷͰɺੵͷ୯Ґݩ 1 ʹ͍ͭͯɺϕ1
    (1) = 1
    ͕੒Γཱͭɻैͬͯɺa1
    ϕ1
    (x) = 0 ʹ x = 1 Λಋೖ͢Δͱɺϕ1
    (1) = 1 ΑΓɺa1
    = 0 ͕ಘΒΕΔɻ
    ࣍ʹɺ
    n−1 ݸͷ৔߹ʹ੒ཱ͢Δ΋ͷͱԾఆͯ͠ɺ
    n ≥ 2 ͷ৔߹Λߟ͑Δͱɺ
    ϕn
    = ϕ1
    ΑΓɺ
    ϕn
    (x0
    ) = ϕ1
    (x0
    )
    ͱͳΔ x0
    ∈ E ͕ଘࡏ͢Δɻ͜ͷ࣌ɺ
    n
    i=1
    ai
    ϕi
    (x) = 0 (9)
    ͷ྆ลʹ ϕn
    (x0
    ) Λֻ͚Δͱɺ
    n
    i=1
    ai
    ϕi
    (x)ϕn
    (x0
    ) = 0 (10)
    ͕੒ཱ͢Δɻ·ͨɺ(9) ͸೚ҙͷ x ∈ E ʹ͍ͭͯ੒ΓཱͭͷͰɺx Λ xx0
    ʹஔ͖׵͑ͨ৔߹Λߟ͑Δͱɺ
    n
    i=1
    ai
    ϕi
    (x)ϕi
    (x0
    ) = 0 (11)
    ͱͳΔɻ͜͜Ͱ͸ɺࣗݾಉܕࣸ૾ͷੑ࣭͔Β ϕi
    (xx0
    ) = ϕi
    (x)ϕi
    (x0
    ) ͕੒Γཱͭ͜ͱΛ༻͍ͨɻ͜͜Ͱɺ(10)
    ͱ (11) ͷลʑΛҾ͘ͱɺi = n ͷ߲͕૬ࡴ͢Δ͜ͱʹ஫ҙͯ͠ɺ
    n−1
    i=1
    ai
    ϕi
    (x) {ϕn
    (x0
    ) − ϕi
    (x0
    )} = 0
    ͕ಘΒΕΔɻैͬͯɺؼೲ๏ͷԾఆΑΓɺ
    ai
    {ϕn
    (x0
    ) − ϕi
    (x0
    )} = 0 (i = 1, · · · , n − 1)
    ͕੒Γཱͭɻ͜͜Ͱɺಛʹ i = 1 ͷ৔߹Λߟ͑Δͱɺϕn
    (x0
    ) − ϕ1
    (x0
    ) = 0 Ͱ͋Δ͜ͱ͔Βɺa1
    = 0 ͕ಘΒΕ
    Δɻैͬͯɺ࠷ॳͷ৚݅͸ɺ
    ∀x ∈ E;
    n
    i=2
    ai
    ϕi
    (x) = 0
    ͱͳΓɺؼೲ๏ͷԾఆΑΓɺai
    = 0 (i = 2, · · · , n) ͕ಘΒΕΔɻ ˙
    ࣍͸ɺ෦෼ମΛݻఆ͢Δࣗݾಉܕࣸ૾ͷݸ਺ʹؔ͢Δิ୊Ͱ͢ɻ
    11

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  12. ิ୊ 1 ମͷ֦େ E/F ʹ͓͍ͯɺF Λݻఆ͢Δ E ͷ૬ҧͳΔࣗݾಉܕࣸ૾ͷݸ਺͸ɺ֦େͷ࣍਺Ͱ͓͑͞
    ΒΕΔɻͭ·Γɺ࣍ͷෆ౳͕ࣜ੒ཱ͢Δ*5ɻ
    |Aut(E/F)| ≤ [E : F]
    ʢূ໌ʣ
    [E : F] = m ͱͯ͠ɺ{α1
    , · · · , αm
    } ⊂ E Λ F ্ͷϕΫτϧۭؒ E ͷجఈͱ͢ΔɻࠓɺAut(E/F) ͷཁ
    ૉɺ͢ͳΘͪɺF Λݻఆ͢Δ E ͷࣗݾಉܕࣸ૾Ͱɺ૬ҧͳΔ΋ͷ͕ n ݸ͋Δͱͯ͠ɺͦΕΒΛ
    Aut(E/F) = {ϕ1
    , · · · , ϕn
    }
    ͱ͢Δɻ͜ͷ࣌ɺn ≤ m ͕੒Γཱͭ͜ͱΛࣔ͢ɻ
    ͸͡ΊʹɺE ͷݩΛ m ݸฒ΂ͨ਺ϕΫτϧۭؒ Em ͷ n ݸͷݩΛ࣍Ͱఆٛ͢Δɻ
    vi
    = (ϕi
    (α1
    ), · · · , ϕi
    (αm
    )) (i = 1, · · · , n)
    ͜ͷ࣌ɺ͜ΕΒͷ਺ϕΫτϧ͸ޓ͍ʹҰ࣍ಠཱͰ͋Δ͜ͱ͕ࣔͤΔɻ࣮ࡍɺ
    n
    i=1
    βi
    vi
    = 0 (β1
    , · · · βn
    ∈ E)
    ͱ͢Δͱɺ਺ϕΫτϧͷ֤੒෼Λॻ͖Լͯ͠ɺ
    n
    i=1
    βi
    ϕi
    (αj
    ) = 0 (j = 1, · · · , m)
    ͕ಘΒΕΔɻैͬͯɺ೚ҙͷ x ∈ E ʹରͯ͠ɺ͜ΕΛجఈ {α1
    , · · · , αm
    } ͷઢܗ݁߹Ͱදͯ͠ɺ
    x =
    m
    j=1
    aj
    αj
    (aj
    ∈ F)
    ͱ͢Δ࣌ɺ࣍ͷؔ܎͕੒ཱ͢Δɻ
    n
    i=1
    βi
    ϕi
    (x) =
    n
    i=1
    βi
    ϕi


    m
    j=1
    aj
    αj

     =
    m
    j=1
    aj
    n
    i=1
    βi
    ϕi
    (αj
    ) = 0
    ͜͜Ͱ͸ɺಉܕࣸ૾ͷੑ࣭ɺ͓Αͼɺϕi
    ͸ F ͷݩ aj
    Λಈ͔͞ͳ͍ͷͰ ϕi
    (aj
    ) = aj
    ͱͳΔࣄΛ༻͍ͯɺ࣍
    ͷมܗΛߦ͍ͬͯΔɻ
    ϕi


    m
    j=1
    aj
    αj

     =
    m
    j=1
    ϕi
    (aj
    )ϕ(αj
    ) =
    m
    j=1
    aj
    ϕ(αj
    )
    ैͬͯɺ
    n
    i=1
    βi
    ϕi
    (x) = 0
    Ͱ͋Γɺఆཧ 4 ΑΓ βi
    = 0 (i = 1, · · · , n) ͕ಘΒΕΔ͕ɺ͜Ε͸ɺ{v1
    , · · · , vn
    } ͕Ұ࣍ಠཱͰ͋ΔࣄΛҙຯ
    ͢ΔɻҰํɺm ࣍ݩͷ਺ϕΫτϧۭؒͰҰ࣍ಠཱͳݩ͸ߴʑ m ݸͳͷͰɺn ≤ m ͕ݴ͑Δɻ ˙
    *5 |A| ͸ɺू߹ A ͷཁૉ਺Λද͠·͢ɻ
    12

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  13. ઌ ΄ Ͳ ͷ ྫ 3-1 ͱ ྫ 3-2 Ͱ ͸ ɺ΍ ΍ ௚ ײ త ͳ ख ๏ ʹ Α Γ ɺΨ ϩ Ξ ܈ Aut(Q(

    2)/Q)ɺ͓ Α ͼ ɺ
    Aut(Q(

    2,

    3)/Q) ͷཁૉͱͳΔࣗݾಉܕࣸ૾Λൃݟ͠·͕ͨ͠ɺิ୊ 1 ʹΑΓɺͦ͜Ͱൃݟͨ͠΋ͷ͕Ψ
    ϩΞ܈ͷ͢΂ͯͷཁૉͰ͋Δ͜ͱ͕อূ͞Ε·͢ɻྫ 3-1 Ͱ͋Ε͹ɺ[Q(

    2) : Q] = 2 Ͱ͋Δ͜ͱ͔ΒɺΨϩ
    Ξ܈ͷཁૉ͸ߴʑ 2 ݸͰɺ{1, ϕ} ͕ͦͷ͢΂ͯͱͳΓ·͢ɻ͋Δ͍͸ɺྫ 3-2 Ͱ͋Ε͹ɺ[Q(

    2,

    3) : Q] = 4
    Ͱ͋Δ͜ͱ͔ΒɺΨϩΞ܈ͷཁૉ͸ߴʑ 4 ݸͰɺ{1, ϕ1
    , ϕ2
    , ϕ3
    } ͕ͦͷ͢΂ͯͱͳΓ·͢ɻ
    ͦΕͰ͸ɺҎ্ͷ४උͷ΋ͱʹɺલड़ͷఆཧΛূ໌͠·͢ɻ
    ఆཧ 5 ମ E ͷ༗ݶͳࣗݾಉܕ෦෼܈ G ⊂ Aut(E) ͕ଘࡏͨ͠৔߹ɺ
    Aut(E/EG) = G
    ͓Αͼɺ
    [E : EG] = |G|
    ͕੒ཱ͢Δɻ
    ʢূ໌ʣ
    G ⊆ Aut(E/EG) ͸ࣗ໌Ͱɺ͜ΕΑΓɺ
    |G| ≤ |Aut(E/EG)| ≤ [E : EG]
    ͕੒ཱ͢Δɻ2 ͭ໨ͷෆ౳ࣜ͸ɺิ୊ 1 ʹΑΔɻैͬͯɺٯ޲͖ͷෆ౳ࣜ |G| ≥ [E : EG] ͕ࣔͤΕ͹ɺ
    |G| = |Aut(E/EG)| = [E : EG] ͱͳΓɺू߹ͱͯ͠ͷཁૉ਺͕౳͍͜͠ͱ͔ΒɺG = Aut(E/EG) ΋ݴ͑
    Δɻͦ͜ͰɺҎԼɺ|G| = m ͱͯ͠ɺ[E : EG] ≤ m Λࣔ͢ɻͦΕʹ͸ɺEG ্ͷϕΫτϧۭؒ E ͔Β n ݸ
    ʢn > mʣͷݩ {α1
    , · · · , αn
    } Λ೚ҙʹऔͬͨ࣌ɺ͜ΕΒ͕Ұ࣍ैଐͳϕΫτϧͰ͋Δ͜ͱ͕ݴ͑Ε͹Α͍ɻ
    ࠓɺm ݸ͋Δ G ͷཁૉΛ۩ମతʹฒ΂ͨ΋ͷΛ
    G = {ϕ1
    , · · · , ϕm
    } (ϕ1
    = 1)
    ͱͯ͠ɺm ࣍ݩͷ਺ϕΫτϧۭؒ Em ͷ n ݸͷݩΛ࣍Ͱఆٛ͢Δɻ
    vi
    = (ϕ1
    (αi
    ), · · · , ϕm
    (αi
    )) (i = 1, ʜ, n)
    ͜ΕΒͷϕΫτϧͰҰ࣍ಠཱͳ΋ͷ͸ߴʑ m ݸʢm < nʣͰ͋Δ͜ͱʹ஫ҙͯ͠ɺ࣮ࡍʹҰ࣍ಠཱͳ΋ͷ͕
    r ݸ͋Δͱͨ͠৔߹ɺॱ൪Λฒ΂ସ͑ͯɺͦΕΒΛ {v1
    , · · · , vr
    } ͱ͢Δɻ͜ͷ࣌ɺvn
    ͸࠷ॳͷ r ݸʹରͯ͠
    Ұ࣍ैଐͱͳΓɺ
    vn
    =
    r
    k=1
    βk
    vk
    (β1
    , · · · , βr
    ∈ E) (12)
    ͱॻ͚Δɻ͜Εʹ vi
    ͷఆٛΛ୅ೖͯ͠ɺ੒෼͝ͱʹදࣔ͢Δͱɺ
    ϕj
    (αn
    ) =
    r
    k=1
    βk
    ϕj
    (αk
    ) (j = 1, · · · , m) (13)
    ͕ಘΒΕΔɻ͞Βʹɺ͜ͷ྆ลʹɺ೚ҙͷ ϕ ∈ G Λ࡞༻ͤ͞Δͱɺϕ ◦ ϕj
    = ϕj′
    ͱͯ͠ɺ
    ϕj′
    (αn
    ) =
    r
    k=1
    ϕ(βk
    )ϕj′
    (αk
    ) (j′ = 1, · · · , m) (14)
    13

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  14. ͕ಘΒΕΔɻӈลΛܭࢉ͢Δࡍ͸ɺࣗݾಉܕࣸ૾ͷੑ࣭Λ༻͍ͯɺ࣍ͷมܗΛߦͬͨɻ
    ϕ (βk
    ϕj
    (αk
    )) = ϕ(βk
    )ϕ (ϕj
    (αk
    )) = ϕ(βk
    )ϕj′
    (αk
    )
    ·ͨɺG = {ϕ1
    , · · · , ϕm
    } ͸༗ݶ܈Ͱ͋Δ͜ͱ͔Βɺϕ ◦ G = {ϕ ◦ ϕ1
    , · · · , ϕ ◦ ϕm
    } ͸ɺ࠶ͼू߹ͱͯ͠ G ʹ
    Ұக͢Δ఺ʹ΋஫ҙ͢Δɻ
    ͜͜Ͱɺ(14) ΛϕΫτϧ vi
    Ͱͷදهʹ໭͢ͱɺ
    vn
    =
    r
    k=1
    ϕ(βk
    )vk
    (15)
    ͱͳΓɺ(12) ͱ (15) Λൺֱ͢Δͱɺ {v1
    , · · · , vr
    } ͸Ұ࣍ಠཱͰ͋Δ͜ͱ͔Βɺ
    ϕ(βk
    ) = βk
    (k = 1, · · · , r)
    ͕ಘΒΕΔɻ͜Ε͸ɺβk
    ͸ G Ͱݻఆ͞ΕΔࣄΛҙຯ͓ͯ͠Γɺ
    β1
    , · · · , βr
    ∈ EG (16)
    ͕ݴ͑Δɻ͞Βʹɺ(13) Ͱ j = 1 ͷ৔߹ΛऔΓग़͢ͱɺϕ1
    = 1 ΑΓɺ
    αn
    =
    r
    k=1
    βk
    αk
    (17)
    ͕ಘΒΕΔɻ(16)(17) ͸ɺαn
    ͸ɺ{α1
    , · · · , αr
    } ʹରͯ͠ɺEG ্ͷϕΫτϧۭؒͰҰ࣍ैଐͰ͋Δ͜ͱΛࣔ
    ͢ɻ ˙
    ྫ 3-3

    ఆཧ 5 Λ༻͍ͯɺ֦େ Q(

    2)/Q ͷΨϩΞ܈ Aut(Q(

    2)/Q) Λݫີʹಋग़ͯ͠Έ·͢ɻ·ͣɺE =
    Q(

    2) ͱͯ͠ɺྫ 3-1 Ͱఆٛͨ͠ G = {1, ϕ} ∼
    = S2
    ͕ɺࣗݾಉܕ܈ Aut(E) ͷ༗ݶ෦෼܈Ͱ͋Δ͜ͱ
    ͸ɺྫ 3-1 ͰߦͬͨΑ͏ʹɺ௚઀ܭࢉͰ֬ೝͰ͖·͢ɻ·ͨɺG ͕ݻఆ͢Δݩͷू߹͸ɺEG = Q ͱͳ
    ΔͷͰɺఆཧ 5 ʹΑΓɺAut(Q(

    2)/Q) = Aut(E/EG) = G ∼
    = S2
    ͕ಘΒΕ·͢ɻ
    ֦େ Q(

    2,

    3)/Q ʹ͍ͭͯ΋ಉ༷ͷٞ࿦͕ՄೳͰ͢ɻE = Q(

    2,

    3) ͱͯ͠ɺྫ 3-2 Ͱఆٛͨ͠
    G = {1, ϕ1
    , ϕ2
    , ϕ3
    } ͸ɺࣗݾಉܕ܈ Aut(E) ͷ༗ݶ෦෼܈Ͱ͋ΓɺG ͕ݻఆ͢Δݩͷू߹͸ɺEG = Q
    ͱͳΓ·͢ɻैͬͯɺఆཧ 5 ʹΑΓɺAut(Q(

    2,

    3)/Q) = Aut(E/EG) = G ͕ಘΒΕ·͢ɻ

    ͜ͷྫ͔Β෼͔ΔΑ͏ʹɺମͷ֦େ E/F ʹ͓͍ͯɺF ͷΈΛݻఆ͢Δ Aut(E) ͷ༗ݶ෦෼܈ Gɺ͢ͳΘ
    ͪɺEG = F Λຬͨ͢Α͏ͳ G ͕ൃݟͰ͖Ε͹ɺͦΕ͕͜ͷ֦େͷΨϩΞ܈ Aut(E/F) Λ༩͑Δ͜ͱʹͳΓ
    ·͢ɻҰൠʹ͸ɺ͜ͷΑ͏ͳ G ͕ඞͣଘࡏ͢Δͱ͍͏Θ͚Ͱ͸͋Γ·ͤΜ͕ɺಛʹ͜ͷΑ͏ͳ G ͕ଘࡏ͢Δ
    ମͷ֦େΛΨϩΞ֦େͱݺͼ·͢ɻઌ΄ͲͷྫͰऔΓ্͛ͨɺQ(

    2)/Qɺ͓ΑͼɺQ(

    2,

    3)/Q ͸ɺͲͪ
    Β΋ΨϩΞ֦େͰ͋Δ͜ͱʹͳΓ·͢ɻ·ͨɺఆཧ 5 ΑΓɺΨϩΞ܈ʹؚ·ΕΔࣗݾಉܕࣸ૾ͷ਺ʹ͍ͭͯɺ
    ࣍ͷܥ͕ಘΒΕ·͢ɻ
    ܥ 2 ମͷ֦େ E/F ͕ΨϩΞ֦େͰ͋Δ࣌ɺ࣍ͷؔ܎͕੒ཱ͢Δɻ
    |Aut(E/F)| = [E : F]
    14

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  15. ʢূ໌ʣ
    E/F ͕ΨϩΞ֦େͰ͋Ε͹ɺG = Aut(E/F) ⊂ Aut(E) ͕ଘࡏͯ͠ɺF = EG ͱͳΔͷͰɺ͜ΕΒͷؔ
    ܎Λఆཧ 5 ͷ݁Ռʹ୅ೖ͢Δ͜ͱͰಘΒΕΔɻ ˙
    3.3 ΨϩΞ֦େͷجຊ৚݅
    ࣍ͷఆཧ͸ɺ֦େ E/F ͕ΨϩΞ֦େͰ͋ΔͨΊͷඞཁे෼৚݅Λ༩͑·͢ɻAut(E/F) ʹؚ·ΕΔࣗݾಉ
    ܕࣸ૾͸ɺF Λݻఆ͢ΔΘ͚Ͱ͕͢ɺF Ҏ֎ͷݩΛݻఆ͢Δ৔߹΋͋Γ·͢ͷͰɺҰൠʹ͸ɺEAut(E/F ) ⊇ F
    ͱ͍͏แؚؔ܎͕੒Γཱͪ·͢ɻ͜ͷ 2 ͭͷू߹͕Ұக͢Δ͜ͱ͕ɺE/F ͕ΨϩΞ֦େͰ͋Δ͜ͱͱಉ஋ʹ
    ͳΓ·͢ɻ
    ఆཧ 6 ମͷ֦େ E/F ͕ΨϩΞ֦େͰ͋Δ͜ͱ͸ɺ
    EAut(E/F ) = F
    ͕੒Γཱͭ͜ͱͱಉ஋Ͱ͋Δɻ
    ʢূ໌ʣ
    े෼৚݅ɿEAut(E/F ) = F ͱԾఆ͢ΔͱɺG = Aut(E/F) ͱͯ͠ɺEG = F ͕੒ཱ͢ΔͷͰɺE/F ͸Ψϩ
    Ξ֦େͰ͋Δɻ
    ඞཁ৚݅ɿAut(E) ͷ༗ݶ෦෼܈ G ʹରͯ͠ɺఆཧ 5 ΑΓ
    G = Aut(E/EG) (18)
    ͕੒ཱ͢ΔɻE/F ͕ΨϩΞ֦େͱ͢ΔͱɺF = EG ͱͳΔ Aut(E) ͷ༗ݶ෦෼܈ G ͕ଘࡏ͢ΔͷͰɺ͜Ε
    ʹ (18) Λ୅ೖͯ͠ɺ
    F = EAut(E/EG) = EAut(E/F )
    ͕੒ཱ͢Δɻ ˙
    ֦େ E/F ͕ΨϩΞ֦େͰ͋Δ৔߹ɺఆཧ 6 ʹΑΓɺG = Aut(E/F) ͱͯ͠ɺEG = F ͕੒Γཱͭ͜ͱ͔
    ΒɺจݙʹΑͬͯ͸ɺΨϩΞ֦େΛදه͢Δࡍʹɺه߸ F Λ༻͍ͣʹɺE/EGɺ͋Δ͍͸ɺE/EAut(E/F) ͷ
    Α͏ʹهࡌ͢Δ͜ͱ͕͋Γ·͢ɻ͜ͷΑ͏ͳදهΛݟͨ৔߹͸ɺಛʹઆ໌͕ͳͯ͘΋ɺΨϩΞ֦େͰ͋Δͱߟ
    ͑Δ͜ͱ͕Ͱ͖·͢ɻ
    4 ΨϩΞཧ࿦ͷجຊఆཧ
    4.1 ΨϩΞ֦େͷதؒମ
    ͜͜Ͱ͸ɺ2 ஈ֊ͷମͷ֦େʹ͍ͭͯߟ͑·͢ɻࠓɺମͷ֦େ M/Fɺ͓ΑͼɺE/M ͕͋ͬͨͱ͢Δͱɺू
    ߹ͱͯ͠ͷแؚؔ܎
    E ⊃ M ⊃ F (19)
    15

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  16. ͕੒Γཱͪɺ͞ΒʹɺE/F ΋ମͷ֦େͱΈͳ͢͜ͱ͕Ͱ͖·͢ɻ͜ͷΑ͏ͳؔ܎Λຬͨ͢ M Λ֦େ E/F ͷ
    தؒମͱݺͼ·͢ɻ͋Δ͍͸ɺ֦େ E/F ʹରͯ͠ɺ(19) Λຬͨ͢ମ M ͷ͜ͱΛதؒମͱఆٛͯ͠΋ಉ͜͡
    ͱʹͳΓ·͢ɻ
    ͦͯ͠ɺ͜ͷ࣌ɺE/F ͕ΨϩΞ֦େͰ͋Ε͹ɺE/M ΋ΨϩΞ֦େʹͳ͍ͬͯΔࣄ͕ূ໌͞Ε·͢ɻΑΓ
    ۩ମతʹ͸ɺG = Aut(E/F) ͷ෦෼܈ H = Aut(E/M) Λ༻͍ͯɺM = EH ͕੒ΓཱͪɺE/M ͕ΨϩΞ
    ֦େͰ͋Δ͜ͱ͕෼͔Γ·͢ɻ͞Βʹ͸ɺ͜Εͱಉ༷ͷؔ܎Λ௨ͯ͠ɺ֦େ E/F ʹର͢Δ͢΂ͯͷதؒମ͕
    G ͷ෦෼܈ͱ 1 ର 1 ʹରԠ͢Δͱ͍͏ஶ͍݁͠Ռ͕ಘΒΕ·͢ɻͭ·ΓɺG ͷߏ଄͔Β E ͱ F ͷؒʹଘࡏ
    ͠ಘΔதؒମ͕ܾ·Γɺ͜ΕʹΑΓɺF ͔Βͷ΂͖֦ࠜେͰ E ͕ಘΒΕΔ͔Ͳ͏͔͕෼͔Γ·͢ɻ͜ΕΛΨ
    ϩΞཧ࿦ͷجຊఆཧͱݺͼɺຊઅͰ͸ɺ͜ͷఆཧΛॱΛ௥ͬͯࣔ͠·͢ɻ
    ·ͣɺ४උͱͯ͠ɺఆཧ 4 ͱิ୊ 1 ʹ͍ͭͯɺ͜ΕΒΛࣗݾಉܕࣸ૾͔Β४ಉܕࣸ૾ʹҰൠԽͨ͠΋ͷΛূ
    ໌͠·͢ɻূ໌ͷྲྀΕͦͷ΋ͷ͸ɺఆཧ 4ɺ͓Αͼɺิ୊ 1 ͱେ͖͘͸มΘΓ·ͤΜɻͳ͓ɺ͜͜Ͱ͸ɺମ M
    ͔Βମ E ΁ͷ؀ͱͯ͠ͷ४ಉܕࣸ૾શମΛ
    Hom(M, E)
    ͱ͍͏ه߸Ͱද͠·͢ɻϕ ∈ Hom(M, E) ͸ɺα1
    , α2
    ∈ M ʹରͯ͠ɺ࣍ͷ৚݅Λຬͨ͢ M ͔Β E ΁ͷࣸ૾
    ʹͳΓ·͢ɻ
    ϕ(α1
    + α2
    ) = ϕ(α1
    ) + ϕ(α2
    )
    ϕ(α1
    α2
    ) = ϕ(α1
    )ϕ(α2
    )
    ࣗݾಉܕࣸ૾ͱҟͳΓɺ४ಉܕࣸ૾͸શ୯ࣹͱ͸ݶΓ·ͤΜɻࣸ૾ͷ߹੒Λఆٛ͢Δ͜ͱ΋Ͱ͖·ͤΜͷͰɺ
    Aut(E) ͷΑ͏ʹɺࣸ૾ͷ߹੒ʹؔͯ͠܈Λߏ੒͢Δ΋ͷͰ΋͋Γ·ͤΜɻͨͩ͠ɺ؀ͱͯ͠ͷ४ಉܕࣸ૾ʹ
    ͓͍ͯɺࠓͷΑ͏ʹࣸ૾ͷఆٛҬ͕ମͰ͋Δ৔߹͸ɺඞͣɺ୯ࣹʹͳΔ͜ͱ͕஌ΒΕ͍ͯ·͢ [4]ɻ͞Βʹɺ
    Hom(M, E) ʹؚ·ΕΔࣸ૾ͷதͰɺಛʹ F ͷݩΛಈ͔͞ͳ͍ࣸ૾ͷશମΛ
    HomF
    (M, E)
    ͱ͍͏ه߸Ͱද͠·͢ɻ
    ఆཧ 7 ʢσσΩϯτͷఆཧʣ
    ମ M ͱମ E ʹ͍ͭͯɺM ͔Β E ΁ͷ૬ҧͳΔ४ಉܕࣸ૾ ϕ1
    , · · · ϕn
    ∈ Hom(M, E) ͕༩͑ΒΕͨ࣌ɺ
    a1
    , · · · , an
    ∈ E ʹ͍ͭͯɺ
    ∀x ∈ M;
    n
    i=1
    ai
    ϕ(x) = 0
    ͕੒ΓཱͭͳΒ͹ɺ
    ai
    = 0 (i = 1, · · · , n)
    Ͱ͋Δɻ
    ʢূ໌ʣ
    n ʹ͍ͭͯͷؼೲ๏Ͱࣔ͢ɻn = 1 ͷ࣌ɺϕ1
    ͸४ಉܕࣸ૾ͳͷͰɺੵͷ୯Ґݩ 1 ʹ͍ͭͯɺϕ1
    (1) = 1 ͕
    ੒Γཱͭɻैͬͯɺa1
    ϕ1
    (x) = 0 ʹ x = 1 Λಋೖ͢Δͱɺϕ1
    (1) = 1 ΑΓɺa1
    = 0 ͕ಘΒΕΔɻ
    16

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  17. ࣍ʹɺ
    n−1 ݸͷ৔߹ʹ੒ཱ͢Δ΋ͷͱԾఆͯ͠ɺ
    n ≥ 2 ͷ৔߹Λߟ͑Δͱɺ
    ϕn
    = ϕ1
    ΑΓɺ
    ϕn
    (x0
    ) = ϕ1
    (x0
    )
    ͱͳΔ x0
    ∈ M ͕ଘࡏ͢Δɻ͜ͷ࣌ɺ
    n
    i=1
    ai
    ϕi
    (x) = 0 (20)
    ͷ྆ลʹ ϕn
    (x0
    ) Λֻ͚Δͱɺ
    n
    i=1
    ai
    ϕi
    (x)ϕn
    (x0
    ) = 0 (21)
    ͕੒ཱ͢Δɻ·ͨɺ(20) ͸೚ҙͷ x ∈ M ʹ͍ͭͯ੒ΓཱͭͷͰɺx Λ xx0
    ʹஔ͖׵͑ͨ৔߹Λߟ͑Δͱɺ
    n
    i=1
    ai
    ϕi
    (x)ϕi
    (x0
    ) = 0 (22)
    ͱͳΔɻ͜͜Ͱ͸ɺ४ಉܕࣸ૾ͷੑ࣭͔Β ϕi
    (xx0
    ) = ϕi
    (x)ϕi
    (x0
    ) ͕੒Γཱͭ͜ͱΛ༻͍ͨɻ͜͜Ͱɺ(21)
    ͱ (22) ͷลʑΛҾ͘ͱɺi = n ͷ߲͕૬ࡴ͢Δ͜ͱʹ஫ҙͯ͠ɺ
    n−1
    i=1
    ai
    ϕi
    (x) {ϕn
    (x0
    ) − ϕi
    (x0
    )} = 0
    ͕ಘΒΕΔɻैͬͯɺؼೲ๏ͷԾఆΑΓɺ
    ai
    {ϕn
    (x0
    ) − ϕi
    (x0
    )} = 0 (i = 1, · · · , n − 1)
    ͕੒Γཱͭɻ͜͜Ͱɺಛʹ i = 1 ͷ৔߹Λߟ͑Δͱɺϕn
    (x0
    ) − ϕ1
    (x0
    ) = 0 Ͱ͋Δ͜ͱ͔Βɺa1
    = 0 ͕ಘΒΕ
    Δɻैͬͯɺ࠷ॳͷ৚݅͸ɺ
    ∀x ∈ M;
    n
    i=2
    ai
    ϕi
    (x) = 0
    ͱͳΓɺؼೲ๏ͷԾఆΑΓɺai
    = 0 (i = 2, · · · , n) ͕ಘΒΕΔɻ ˙
    ิ୊ 2 ମͷ֦େ E/F ͷ೚ҙͷதؒମ M ʹରͯ͠ɺF Λݻఆ͢Δ M ͔Β E ͷ૬ҧͳΔ४ಉܕࣸ૾ͷݸ
    ਺͸ɺ֦େ M/F ͷ࣍਺Ͱ͓͑͞ΒΕΔɻͭ·Γɺ࣍ͷෆ౳͕ࣜ੒ཱ͢Δɻ
    |HomF
    (M, E)| ≤ [M : F]
    ʢূ໌ʣ
    [M : F] = m ͱͯ͠ɺ{α1
    , · · · , αm
    } ⊂ M Λ F ্ͷϕΫτϧۭؒ M ͷجఈͱ͢ΔɻࠓɺF Λݻఆ͢Δ
    M ͔Β E ͷ४ಉܕࣸ૾Ͱɺ૬ҧͳΔ΋ͷ͕ n ݸ͋Δͱͯ͠ɺͦΕΒΛ
    HomF
    (M, E) = {ϕ1
    , · · · , ϕn
    }
    ͱ͢Δɻ͜ͷ࣌ɺn ≤ m ͕੒Γཱͭ͜ͱΛࣔ͢ɻ
    ͸͡ΊʹɺE ͷݩΛ m ݸฒ΂ͨ਺ϕΫτϧۭؒ Em ͷ n ݸͷݩΛ࣍Ͱఆٛ͢Δɻ
    vi
    = (ϕi
    (α1
    ), · · · , ϕi
    (αm
    )) (i = 1, · · · , n)
    ͜ͷ࣌ɺ͜ΕΒͷ਺ϕΫτϧ͸ޓ͍ʹҰ࣍ಠཱͰ͋Δ͜ͱ͕ࣔͤΔɻ࣮ࡍɺ
    n
    i=1
    βi
    vi
    = 0 (β1
    , · · · βn
    ∈ E)
    17

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  18. ͱ͢Δͱɺ਺ϕΫτϧͷ֤੒෼Λॻ͖Լͯ͠ɺ
    n
    i=1
    βi
    ϕi
    (αj
    ) = 0 (j = 1, · · · , m)
    ͕ಘΒΕΔɻैͬͯɺ೚ҙͷ x ∈ M ʹରͯ͠ɺ͜ΕΛجఈ {α1
    , · · · , αm
    } ͷઢܗ݁߹Ͱදͯ͠ɺ
    x =
    m
    j=1
    aj
    αj
    (aj
    ∈ F)
    ͱ͢Δ࣌ɺ࣍ͷؔ܎͕੒ཱ͢Δɻ
    n
    i=1
    βi
    ϕi
    (x) =
    n
    i=1
    βi
    ϕi


    m
    j=1
    aj
    αj

     =
    m
    j=1
    aj
    n
    i=1
    βi
    ϕi
    (αj
    ) = 0
    ͜͜Ͱ͸ɺ४ಉܕࣸ૾ͷੑ࣭ɺ͓Αͼɺϕi
    ͸ F ͷݩ aj
    Λಈ͔͞ͳ͍ͷͰ ϕi
    (aj
    ) = aj
    ͱͳΔࣄΛ༻͍ͯɺ
    ࣍ͷมܗΛߦ͍ͬͯΔɻ
    ϕi


    m
    j=1
    aj
    αj

     =
    m
    j=1
    ϕi
    (aj
    )ϕ(αj
    ) =
    m
    j=1
    aj
    ϕ(αj
    )
    ैͬͯɺ
    n
    i=1
    βi
    ϕi
    (x) = 0
    Ͱ͋Γɺఆཧ 7 ΑΓ βi
    = 0 (i = 1, · · · , n) ͕ಘΒΕΔ͕ɺ͜Ε͸ɺ{v1
    , · · · , vn
    } ͕Ұ࣍ಠཱͰ͋ΔࣄΛҙຯ
    ͢ΔɻҰํɺm ࣍ݩͷ਺ϕΫτϧۭؒͰҰ࣍ಠཱͳݩ͸ߴʑ m ͳͷͰɺn ≤ m ͕ݴ͑Δɻ ˙
    ଓ͍ͯɺ֦େ E/F ͷதؒମ M ͕ଘࡏ͢Δͱͯ͠ɺ֦େ E/F ʹର͢ΔΨϩΞ܈ɺ͢ͳΘͪɺF Λݻఆ͢
    Δ E ͷࣗݾಉܕ܈
    G = Aut(E/F)
    ͱɺ֦େ E/M ʹର͢ΔΨϩΞ܈ɺ͢ͳΘͪɺM Λݻఆ͢Δ E ͷࣗݾಉܕ܈
    H = Aut(E/M)
    Λߟ͑·͢ɻM ⊃ F ͱ͍͏แؚؔ܎Λߟ͑ΔͱɺM Λݻఆ͢Δࣸ૾͸ɺඞͣ F ΋ݻఆ͢ΔͷͰɺG ⊇ H
    ͱ͍͏แؚؔ܎͕੒ΓཱͪɺH ͸ G ͷ෦෼܈ʹͳΔ͜ͱ͕෼͔Γ·͢ɻैͬͯɺ܈ͷࠨ৒༨ྨ G/H Λߟ͑
    Δ͜ͱ͕Ͱ͖·͢*6ɻࠨ৒༨ྨ G/H ͱ͍͏ͷ͸ɺG ͷཁૉ σ1
    ͱ σ2
    ʹରͯ͠ɺ
    σ1
    ∼ σ2
    ⇔ ∃ϕ ∈ H; σ1
    = σ2
    ◦ ϕ
    ΋͘͠͸ɺ͜Εͱಉ͡ࣄͰ͕͢ɺ
    σ1
    ∼ σ2
    ⇔ σ−1
    2
    ◦ σ1
    ∈ H
    ͱ͍͏৚݅Ͱಉ஋ؔ܎Λఆٛͯ͠ɺ্هͷؔ܎Λຬͨ͢ σ1
    ͱ σ2
    ΛಉҰࢹͯ͠ಘΒΕΔू߹ʹͳΓ·͢ɻ
    ͦΕͰ͸ɺ͜ͷಉ஋ؔ܎ʹ͸ͲͷΑ͏ͳಛ௃͕͋ΔͷͰ͠ΐ͏͔ʁ σ1
    = σ2
    ◦ ϕ ͱ͍͏ؔ܎͕੒Γཱͭ৔
    ߹ɺ౰વͳ͕Βɺσ1
    ͱ σ2
    ͸ҟͳΔࣸ૾ʹͳΓ·͕͢ɺࠓͷ৔߹ɺH ͷཁૉͰ͋Δ ϕ ͸ɺM ͷݩ͸ಈ͔͞ͳ
    *6 H ͸ਖ਼ن෦෼܈ͱ͸ݶΒͳ͍ͷͰɺG/H ͕܈ʹͳΔͱ͸ݶΓ·ͤΜɻ
    18

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  19. ͍ͱ͍͏ಛ௃͕͋Γ·͢ɻैͬͯɺσ1
    ͱ σ2
    ͸ɺগͳ͘ͱ΋ɺx ∈ M ʹରͯ͠͸ɺσ1
    (x) = σ2
    (x) ͱ͍͏ؔ
    ܎͕੒Γཱͭ͸ͣͰ͢ɻ͜ͷࣄ࣮͸ɺ
    σ1
    |M
    = σ2
    |M
    ͱදݱ͢Δ͜ͱ͕ՄೳͰ͢*7ɻ͜͜·Ͱͷٞ࿦Ͱ͸ɺ͜Ε͸ɺ͋͘·Ͱ΋ σ1
    ∼ σ2
    ͱͳΔͨΊͷඞཁ৚݅ʹ
    ͗͢·ͤΜ͕ɺ࣮͸ɺ͜Ε͸े෼৚݅Ͱ΋͋Γɺ࣍ͷಉ஋ؔ܎͕੒Γཱͪ·͢ɻ
    σ1
    |M
    = σ2
    |M
    ⇔ σ−1
    2
    ◦ σ1
    ∈ H
    ͢͜͠ݴ͍ํΛม͑ΔͱɺG ͷཁૉʹରͯ͠ɺM ͷ૾͕Ұக͢Δ΋ͷΛಉҰࢹ͢Δ͜ͱͱɺ৒༨ྨ G/H ͷ
    ҙຯͰಉҰࢹ͢Δ͜ͱ͸ɺಉ͜͡ͱʹͳΓ·͢ɻ͜ͷࣄ࣮Λಉ஋ྨͷؒͷࣸ૾ͱͯ͠දݱͨ͠΋ͷ͕࣍ͷิ୊
    ʹͳΓ·͢ɻ
    ิ୊ 3 ମͷ֦େ E/F ͷதؒମ M ͕ଘࡏ͢Δ࣌ɺ
    G = Aut(E/F), H = Aut(E/M)
    ͱ͢ΔͱɺH ͸ G ͷ෦෼܈ͱͳΓɺࠨ৒༨ྨ G/H Λߟ͑Δ͜ͱ͕Ͱ͖ΔɻҰํɺG ͷఆٛҬΛ M ʹ੍ݶ
    ͯ͠ಘΒΕΔ M ͔Β E ΁ͷ४ಉܕࣸ૾ͷू߹Λ
    S = {σ|M
    | σ ∈ G}
    ͱ͢Δɻ͜Ε͸ɺG ͷཁૉͰ M ͷ૾͕Ұக͢Δ΋ͷΛ M ͔Β E ΁ͷࣸ૾ͱͯ͠ಉҰࢹͨ͠ू߹Ͱ͋Δɻ
    ͜ͷ࣌ɺ࣍ͷࣸ૾͸ well-definedɺ͔ͭɺશ୯ࣹͱͳΓɺG/H ͱ S ͷ 1 ର 1 ରԠΛ༩͑Δɻ
    G/H −→ S
    σ −→ σ|M
    ͜͜ʹɺσ ∈ G ͸ɺG/H ͷཁૉΛදΘ͢೚ҙͷ୅දݩͱ͢Δɻ
    ʢূ໌ʣ
    σ′|M
    ∈ Sɺ͓Αͼɺσ ∈ G ʹରͯ͠ɺ߹੒ࣸ૾ (σ ◦ σ′)|M
    Λߟ͑Δͱɺ͜Ε͸࠶ͼ S ͷཁૉʹͳ͍ͬͯ
    Δɻैͬͯɺू߹ S ʹର͢Δ܈ G ͷ࡞༻Λ࣍Ͱఆٛ͢Δ͜ͱ͕Ͱ͖Δɻ
    σ ∈ G : S −→ S
    ϕ = σ′|M
    −→ σ(ϕ) = (σ ◦ σ′)|M
    ͜ͷ࣌ɺఆ͔ٛΒ໌Β͔ͳΑ͏ʹɺ೚ҙͷ ϕ ∈ Sɺ͓Αͼɺ೚ҙͷ σ1
    , σ2
    ∈ G ʹରͯ͠ɺ
    σ1
    (σ2
    (ϕ)) = σ1
    ◦ σ2
    (ϕ) (23)
    ͱ͍͏ਪҠؔ܎͕੒ཱ͢Δɻ
    ͜͜ͰɺG ͷ୯Ґݩɺ͢ͳΘͪɺ߃౳ࣸ૾Λ 1 ͱͯ͠ɺe = 1|M
    ͱ͢Δ࣌ɺ্هͷ࡞༻ʹؔͯ͠ɺe ∈ S Λ
    ݻఆ͢Δ G ͷཁૉΛߟ͑Δͱɺ೚ҙͷ σ ∈ G ʹରͯ͠ɺ
    σ(e) = (σ ◦ 1)|M
    = σ|M
    (24)
    *7 σ|M
    ͸ࣸ૾ σ ͷఆٛҬΛ M ʹ੍ݶͨ͠΋ͷΛද͠·͢ɻ
    19

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  20. Ͱ͋Δ͜ͱ͔Βɺ
    σ(e) = e ⇔ σ|M
    = 1|M
    ͱ͍͏৚͕݅ಘΒΕΔɻ͜Ε͸ɺσ ͕ M ͷݩΛಈ͔͞ͳ͍ࣸ૾ɺ͢ͳΘͪɺH = Aut(E/M) ͷཁૉͰ͋Δ
    ͜ͱͱಉ஋Ͱ͋Δɻ
    σ(e) = e ⇔ σ ∈ H (25)
    Ҏ্ͷ४උͷ΋ͱʹɺू߹ S ͱࠨ৒༨ྨ G/H ͷؒʹ 1 ର 1 ͷରԠؔ܎Λߏ੒͢Δɻࠓɺσ1
    , σ2
    ∈ G ͕ू
    ߹ S ͷಉ͡ཁૉʹରԠ͢Δɺ͢ͳΘͪɺ
    σ1
    |M
    = σ2
    |M
    ͕੒Γཱͭͱ͢Δͱɺ(24) ΑΓɺ
    σ1
    (e) = σ2
    (e)
    ͕ಘΒΕΔɻ͜ͷ྆ลʹ σ−1
    2
    ∈ G Λ࡞༻͢Δͱɺ(23) ΑΓɺ
    (σ−1
    2
    ◦ σ1
    )(e) = e
    ͱͳΓɺ(25) ΑΓɺ
    σ−1
    2
    ◦ σ1
    ∈ H
    ͕ಘΒΕΔɻैͬͯɺσ1
    ͱ σ2
    ͸ G/H ͷಉ͡ཁૉʹରԠ͓ͯ͠Γɺࣸ૾
    S −→ G/H
    σ|M
    −→ σ
    ͸ well-defined ͱͳΔɻ·ͨɺ্هͷٞ࿦ΛٯʹͨͲΔ͜ͱʹΑΓɺσ−1
    2
    ◦ σ1
    ∈ H ͔Β σ1
    |M
    = σ2
    |M
    Λࣔ
    ͢͜ͱ΋Ͱ͖ΔͷͰɺٯ޲͖ͷࣸ૾
    G/H −→ S
    σ −→ σ|M
    ΋ well-defined ͱͳΔɻैͬͯɺ͜ΕΒͷࣸ૾ʹΑͬͯɺS ͱ G/H ͷ 1 ର 1 ରԠ͕ಘΒΕΔɻ ˙
    ͜ΕͰΑ͏΍͘ɺຊઅͷ๯಄ʹ঺հͨ͠ΨϩΞཧ࿦ͷجຊఆཧΛূ໌͢Δ͜ͱ͕Ͱ͖·͢ɻ·ͣ͸ɺ֦େ
    E/M ͕ΨϩΞ֦େʹͳΔࣄΛࣔ͠·͢ɻ
    ఆཧ 8 ʢΨϩΞཧ࿦ͷجຊఆཧʣ
    ΨϩΞ֦େ E/F ʹ͓͚Δ೚ҙͷதؒମ M ʹରͯ͠ɺ
    M = EAut(E/M)
    ͕੒ཱ͢Δɻ͢ͳΘͪɺ֦େ E/M ͸ΨϩΞ֦େͰ͋Δɻ
    ূ໌
    G = Aut(E/F), H = Aut(E/M), S = {σ|M
    | σ ∈ G} ͱ͢Δ࣌ɺิ୊ 3 ΑΓɺG/H ͱ S ʹ 1 ର 1 ରԠ
    ͕ଘࡏͯ͠ɺ
    |G/H| = |S|
    20

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  21. ͕੒ཱ͢ΔɻҰํɺ܈ͷ৒༨ྨͷੑ࣭ΑΓɺ|G/H| ʹ͍ͭͯɺ
    |G/H| =
    |G|
    |H|
    ͱͳΓɺ͞Βʹɺఆཧ 5ɺఆཧ 3 ΑΓɺ
    |G|
    |H|
    =
    [E : EG]
    [E : EH]
    = [EH : EG]
    ͱͳΔͷͰɺ݁ہɺ
    |S| = [EH : EG] = [EH : F] (26)
    ͕ಘΒΕΔɻ࠷ޙͷ౳߸͸ɺE/F ͕ΨϩΞ֦େͰ͋Δ͜ͱ͔Βɺఆཧ 6 ΑΓ EG = F ͱͳΔࣄΛ༻͍ͨɻ
    ͞ΒʹɺS ͷཁૉ͸ɺM ͔Β E ΁ͷ४ಉܕࣸ૾ͰɺF ͷݩΛಈ͔͞ͳ͍ͱ͍͏ੑ࣭Λຬ͍ͨͯ͠ΔͷͰɺ
    S ⊆ HomF
    (M, E) Ͱ͋Γɺิ୊ 2 ΑΓɺ
    |S| ≤ |HomF
    (M, E)| ≤ [M : F] (27)
    ͕੒Γཱͭɻ(26) ͱ (27) Λ͋ΘͤΔͱɺ
    [EH : F] ≤ [M : F] (28)
    ͕ಘΒΕΔɻҰํɺH = Aut(E/M) Ͱ͋Δ͜ͱ͔ΒɺM ͷݩ͸ඞͣ H Ͱݻఆ͞ΕͯɺEH ⊇ M ͱ͍͏แ
    ؚؔ܎͕੒Γཱͭɻͭ·ΓɺEH ͱ M ΛͦΕͧΕ F ্ͷϕΫτϧۭؒͱߟ͑ͨ৔߹ɺM ͸ EH ͷ෦෼ϕ
    ΫτϧۭؒͰ͋Γɺ
    [EH : F] ≥ [M : F] (29)
    ͕੒ཱ͢Δɻ(28) ͱ (29) Λ͋ΘͤΔͱɺ
    [EH : F] = [M : F]
    ͕ಘΒΕΔɻ͜Ε͸ɺM ͸ EH ͷ෦෼ϕΫτϧۭؒͰɺ͔ͭɺϕΫτϧۭؒͱͯ͠ͷ࣍ݩ͕౳͍͜͠ͱΛද
    ͓ͯ͠Γɺ݁ہɺ͜ͷ 2 ͭ͸ಉҰͰɺ
    M = EH = EAut(E/M)
    ͕੒ཱ͢Δɻ ˙
    ఆཧ 8 ͸ɺΨϩΞ֦େ E/F ͷ೚ҙͷதؒମ M ʹରͯ͠ɺG = Aut(E/F) ͷ෦෼܈ H = Aut(E/M) ͕
    ରԠ͚ͮΒΕͯɺM = EH ͕੒Γཱͭ͜ͱΛ͍ࣔͯ͠·͢ɻ࣮͸ɺ͜Εͱ͸ٯʹɺG ͷ೚ҙͷ෦෼܈ H ʹ
    ରͯ͠ɺதؒମ M = EH ΛҰҙʹରԠ͚ͮΔ͜ͱ΋Ͱ͖ͯɺ݁ہɺ෦෼܈ H ͱதؒମ M ͕ 1 ର 1 ʹରԠ
    ͢Δ͜ͱʹͳΓ·͢ɻ͜ΕΛࣔ͢ͷ͕ɺ࣍ͷఆཧͱͳΓ·͢ɻ
    ఆཧ 9 ʢΨϩΞཧ࿦ͷجຊఆཧʣ
    ΨϩΞ֦େ E/F ʹ͓͍ͯɺ͢΂ͯͷதؒମͷू߹Λ FɺΨϩΞ܈ Aut(E/F) ͷ͢΂ͯͷ෦෼܈ͷू߹Λ
    G ͱ͢Δ࣌ɺ࣍͸શ୯ࣹͷࣸ૾Λ༩͑Δɻ
    F −→ G
    M −→ Aut(E/M)
    21

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  22. ͜ͷࣸ૾ͷٯࣸ૾͸ɺ࣍Ͱ༩͑ΒΕΔɻ
    G −→ F
    H −→ EH
    ʢূ໌ʣ
    ͸͡ΊʹɺF −→ G ͷࣸ૾͕શ୯ࣹͰ͋Δ͜ͱΛࣔ͢ɻ·ͣɺ೚ҙͷ H ∈ G ʹରͯ͠ɺM = EH ΛͱΔ
    ͱɺఆཧ 5 ΑΓɺ
    Aut(E/M) = Aut(E/EH) = H
    ͱͳΔͷͰɺ͜ͷࣸ૾͸શࣹͰ͋Δɻ࣍ʹɺM1
    , M2
    ∈ F ʹରͯ͠ɺ
    Aut(E/M1
    ) = Aut(E/M2
    )
    ͕੒ཱ͢Δ৔߹ɺఆཧ 8 ΑΓɺ
    M1
    = EAut(E/M1), M2
    = EAut(E/M2)
    Ͱ͋Δ͜ͱ͔ΒɺM1
    = M2
    ͕੒Γཱͭɻैͬͯɺ͜ͷࣸ૾͸୯ࣹͰ͋ΓɺҎ্ʹΑΓɺશ୯ࣹͰ͋Δ͜ͱ͕
    ࣔ͞Εͨɻ·ͨɺH = Aut(E/M) ͱ͢Δ࣌ɺఆཧ 8 ΑΓɺEH = EAut(E/M) = M ͱͳΔ͜ͱ͔Βɺٯࣸ૾
    G −→ F ʹؔ͢Δओு΋੒Γཱ͍ͬͯΔɻ ˙
    ྫ 4-1

    ྫ 3-3 ͰݟͨΑ͏ʹɺQ(

    2,

    3)/Q ͸ΨϩΞ֦େͰ͋ΓɺͦͷΨϩΞ܈͸ɺྫ 3-2 Ͱఆٛͨ͠ G =
    {1, ϕ1
    , ϕ2
    , ϕ3
    } Ͱ༩͑ΒΕ·͢ɻG ͷ෦෼܈͸ɺ{1, ϕ1
    , ϕ2
    , ϕ3
    }, {1, ϕ1
    }, {1, ϕ2
    }, {1, ϕ3
    }, {1} ͷ 5 ͭͰ
    ͕͢ɺͦΕͧΕʹରԠ͢Δதؒମɺ͢ͳΘͪɺͦΕͧΕͷ෦෼܈͕ݻఆ͢Δݩͷू߹͸ɺ࣍ͷΑ͏ʹܾ·
    Γ·͢ɻ
    • ෦෼܈ {1, ϕ1
    , ϕ2
    , ϕ3
    } ⇐⇒ தؒମ Q
    • ෦෼܈ {1, ϕ1
    } ⇐⇒ தؒମ Q(

    3)
    • ෦෼܈ {1, ϕ2
    } ⇐⇒ தؒମ Q(

    2)
    • ෦෼܈ {1, ϕ3
    } ⇐⇒ தؒମ Q(

    6)
    • ෦෼܈ {1} ⇐⇒ தؒମ Q(

    2,

    3)
    ΨϩΞཧ࿦ͷجຊఆཧʹΑΓɺQ(

    2,

    3)/Q ͷதؒମ͸ɺ্ͷ 5 ͕ͭ͢΂ͯʹͳΓ·͢ɻ

    4.2 தؒମ΁ͷ֦େ͕ΨϩΞ֦େʹͳΔ৚݅
    લઅͷٞ࿦Ͱ͸ɺΨϩΞ֦େ E/F ͷ೚ҙͷதؒମ M ʹରͯ͠ɺ֦େ E/M ͸ΨϩΞ֦େʹͳΔ͜ͱ͕ࣔ
    ͞Ε·ͨ͠ɻͦΕͰ͸ɺ΋͏Ұํͷ֦େ M/F ʹ͍ͭͯɺͪ͜Β͸ΨϩΞ֦େʹͳΔͱݴ͑ΔͷͰ͠ΐ͏͔ʁ
    ࣮͸ɺͪ͜Β͕ΨϩΞ֦େʹͳΔʹ͸ɺҰఆͷ৚͕݅ඞཁͱͳΓ·͢ɻຊઅͰ͸ɺͦͷͨΊͷɺ͍͔ͭ͘ͷಉ
    ஋ͳ৚݅Λࣔ͠·͢ɻͦͷ४උͱͯ͠ɺ2 ͭͷิ୊Λࣔ͠·͢ɻ
    22

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  23. ·ͣɺఆཧ 8 ͷূ໌ͷதͰɺG = Aut(E/F) ͱ͢Δ࣌ɺS = {σ|M
    | σ ∈ G} ʹରͯ͠ɺS ⊆ HomF
    (M, E)
    ͱ͍͏แؚؔ܎͕੒Γཱͭͱ͍͏ࣄ࣮Λར༻͠·ͨ͠ɻ࣮͸ɺ͜ΕΒͷू߹͸Ұக͢Δͱ͍͏ͷ͕ɺ࣍ͷิ୊
    ʹͳΓ·͢ɻ
    ิ୊ 4 ΨϩΞ֦େ E/F ͷ೚ҙͷதؒମ M ʹରͯ͠ɺ
    G = Aut(E/F)
    S = {σ|M
    | σ ∈ G}
    ͱ͢Δ࣌ɺ࣍ͷ 2 ͭͷؔ܎͕੒ཱ͢Δɻ
    S = HomF
    (M, E)
    |S| = |HomF
    (M, E)| = [M : F]
    ʢূ໌ʣ
    ఆཧ 8 ͷূ໌ͷ (26) (27) (29) Ͱ͸ɺH = Aut(E/M) ͱͯ͠ɺ࣍ͷ 3 ͭͷؔ܎Λಋ͍ͨɻ
    |S| = [EH : F]
    |S| ≤ |HomF
    (M, E)| ≤ [M : F]
    [EH : F] ≥ [M : F]
    ͜ΕΒΛ͋ΘͤΔͱɺ
    [EH : F] = |S| ≤ |HomF
    (M, E)| ≤ [M : F] ≤ [EH : F]
    ͱͳΓɺ͜ΕΑΓɺ
    |S| = |HomF
    (M, E)| = [M : F]
    ͕ಘΒΕΔɻS ͷఆٛΑΓɺS ⊆ HomF
    (M, E) ͱ͍͏แؚؔ܎͕੒Γཱ͕ͭɺ্هલ൒ͷ౳ࣜ͸ɺS ͱ
    HomF
    (M, E) ͷཁૉ਺͕౳͍͜͠ͱΛ͍ࣔͯ͠ΔͷͰɺ
    S = HomF
    (M, E)
    ͕੒ཱ͢Δɻ ˙
    ଓ͍ͯɺࠨ৒༨ྨ G/H ͕܈ߏ଄Λ͔࣋ͭͲ͏͔ʹؔΘΔิ୊Λࣔ͠·͢ɻલઅͰٞ࿦ͨ͠Α͏ʹɺΨϩΞ
    ֦େ E/F ͷதؒମ M ʹ͓͍ͯɺ
    G = Aut(E/F), H = Aut(E/M)
    ͱ͢ΔͱɺH ͸ G ͷ෦෼܈ͱͳΓɺࠨ৒༨ྨ G/H Λఆٛ͢Δ͜ͱ͕Ͱ͖·ͨ͠ɻͦͯ͠ɺҰൠʹɺ܈ G ͷ
    ෦෼܈ H ͕
    ∀σ ∈ G; H = σHσ−1
    ͱ͍͏৚݅Λຬͨ࣌͢ɺH ͸ਖ਼ن෦෼܈ͱݺ͹ΕɺG/H ʹରͯࣗ͠વͳ܈ߏ଄͕ಋೖ͞ΕΔ͜ͱ͕஌ΒΕͯ
    ͍·͢ɻ͜ͷ࣌ʹಘΒΕΔ܈ G/H Λ৒༨܈ͱݺͼ·͢ɻࠓͷ৔߹ɺH ͸ඞͣ͠΋ਖ਼ن෦෼܈ʹ͸ͳΓ·ͤ
    Μ͕ɺ͜Εʹ͍ۙ৚͕݅੒Γཱͭ͜ͱΛࣔ͢ิ୊͕࣍ʹͳΓ·͢ɻ
    23

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  24. ิ୊ 5 ΨϩΞ֦େ E/F ͷ೚ҙͷதؒମ M ʹରͯ͠ɺ
    G = Aut(E/F)
    ͱͯ͠ɺ࣍ͷؔ܎͕੒ཱ͢Δɻ
    ∀σ ∈ G; Aut (E/σ(M)) = σAut(E/M)σ−1
    ʢূ໌ʣ
    σ ∈ G Λ 1 ͭݻఆͨ࣌͠ʹɺ͋Δ τ ∈ Aut(E) ͕ Aut (E/σ(M)) ʹଐ͢Δ৚݅ɺ͢ͳΘͪɺσ(M) ͷݩΛ
    ಈ͔͞ͳ͍ͱ͍͏৚݅Λߟ͑Δͱɺ࣍ͷಉ஋มܗ͕ಘΒΕΔɻ
    τ ∈ Aut (E/σ(M)) ⇐⇒ ∀x ∈ M; τ(σ(x)) = σ(x)
    ⇐⇒ ∀x ∈ M; τ ◦ σ(x) = σ(x)
    ⇐⇒ ∀x ∈ M; σ−1 ◦ τ ◦ σ(x) = x
    ⇐⇒ σ−1 ◦ τ ◦ σ ∈ Aut(E/M)
    ⇐⇒ τ ∈ σAut(E/M)σ−1
    ͜Ε͸ɺAut (E/σ(M)) ͱ σAut(E/M)σ−1 ͕ू߹ͱͯ͠Ұக͢Δ͜ͱΛ͓ࣔͯ͠Γɺ͞Βʹɺ೚ҙͷ σ ∈ G
    ʹ͍ͭͯ͜Ε͕੒ཱ͢Δ͜ͱ͔Βɺ
    ∀σ ∈ G; Aut (E/σ(M)) = σAut(E/M)σ−1
    ͕ಘΒΕΔɻ ˙
    ͜ΕͰɺຊઅͷ๯಄Ͱٞ࿦ͨ͠ɺ֦େ M/F ͕ΨϩΞ֦େʹͳΔͨΊͷ৚݅Λࣔ͢४උ͕Ͱ͖·ͨ͠ɻิ୊
    5 ͷ݁ՌΛ༻͍Δͱɺ
    ∀σ ∈ G; σ(M) = M
    ͸ɺH ͕ਖ਼ن෦෼܈ʹͳΔͨΊͷे෼৚݅Ͱ͋Δ͜ͱ͕෼͔Γ·͢ɻ࣮͸ɺ͜Ε͸ඞཁे෼৚݅Ͱ͋Γɺ͞Β
    ʹ͸ɺM/F ͕ΨϩΞ֦େʹͳΔͨΊͷඞཁे෼৚݅Ͱ΋͋Δ͜ͱ͕ɺ࣍ͷఆཧͰࣔ͞Ε·͢ɻ
    ఆཧ 10 ΨϩΞ֦େ E/F ͷ೚ҙͷதؒମ M ʹରͯ͠ɺ
    G = Aut(E/F)
    H = Aut(E/M)
    ͱ͢Δ࣌ɺ࣍ͷ 3 ͭ͸ɺ͢΂ͯɺ֦େ M/F ͕ΨϩΞ֦େͰ͋Δ͜ͱͱಉ஋ͳ৚݅ͱͳΔɻ
    (a) H ͸ G ͷਖ਼ن෦෼܈Ͱ͋Δɻ
    (b) ∀σ ∈ G; σ(M) = M
    (c) S = {σ|M
    | σ ∈ G} ͱͯ͠ɺS = Aut(M/F)
    ʢূ໌ʣ
    ɾ(c) ⇒ M/F ͕ΨϩΞ֦େ
    E/F ͸ΨϩΞ֦େͳͷͰɺఆཧ 6 ΑΓɺ
    EG = F
    24

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  25. ͕੒Γཱͭɻ͜Ε͸ͭ·Γɺ
    ∀σ ∈ G; σ(x) = x (30)
    Λຬͨ͢ x ∈ E ͸ɺF ͷݩͷΈʹݶΒΕΔ͜ͱΛҙຯ͢Δɻ࣍ʹɺMAut(M/F ) Λߟ͑Δͱɺ͜Ε͸ɺM ͷ
    ݩͷதͰɺAut(M/F) ʹΑͬͯಈ͔ͳ͍΋ͷΛදΘ͢ɻแؚؔ܎ M ⊃ F Λߟ͑ΔͱɺAut(M/F) ͸ɺগͳ
    ͘ͱ΋ F ͷݩ͸ಈ͔͞ͳ͍ͷͰɺҰൠʹɺ
    MAut(M/F ) ⊇ F (31)
    ͱ͍͏ؔ܎͕੒ཱ͢Δɻ
    Ұํɺ(c) ΑΓɺ೚ҙͷ σ ∈ G ʹରͯ͠ɺσ|M
    = τ ͱͳΔ τ ∈ Aut(M/F) ͕ଘࡏ͢Δɻैͬͯɺ
    x ∈ MAut(M/F ) ʹ͍ͭͯɺ೚ҙͷ σ ∈ G ʹରͯ͠ɺରԠ͢Δ τ ∈ Aut(M/F) Λ༻͍ͯɺ
    σ(x) = σ|M
    (x) = τ(x) = x
    ͱ͍͏ܭࢉ͕੒ཱ͢Δɻ
    ʢ1 ͭ໨ͷ౳߸͸ɺMAut(M/F ) ⊆ M ΑΓ੒ཱͯ͠ɺ࠷ޙͷ౳߸͸ɺx ∈ MAut(M/F )
    ͔ͭ τ ∈ Aut(M/F) ͱ͍͏લఏΑΓࣗ໌ʹ੒Γཱͭɻ
    ʣ͜Ε͸ɺx ʹରͯ͠ (30) ͕੒ཱ͢Δ͜ͱΛ͓ࣔͯ͠
    Γɺ݁ہɺx ∈ F ͱ͍͏͜ͱʹͳΔɻ͜Ε͕೚ҙͷ x ∈ MAut(M/F ) ʹ͍ͭͯ੒Γཱͭ͜ͱ͔Βɺ
    MAut(M/F ) ⊆ F (32)
    ͕ಘΒΕΔɻ(31) (32) ΑΓɺ
    MAut(M/F ) = F
    ͱͳΓɺఆཧ 6 ʹΑΓɺM/F ͸ΨϩΞ֦େͰ͋Δɻ
    ɾM/F ͕ΨϩΞ֦େ ⇒ (c)
    σ ∈ G ͸ɺF ͷݩΛಈ͔͞ͳ͍ͷͰɺఆٛҬΛ M ʹ੍ݶͯ͠΋ɺ΍͸ΓɺF ͷݩΛಈ͔͢͜ͱ͸ͳ͍ɻ
    ैͬͯɺҰൠʹɺ
    S ⊆ Aut(M/F) (33)
    ͕੒ཱ͢ΔɻҰํɺM/F ͕ΨϩΞ֦େͰ͋Ε͹ɺܥ 2 ΑΓɺ
    [M : F] = |Aut(M/F)|
    ͕੒ཱ͢Δɻैͬͯɺิ୊ 4 ͱ͋Θͤͯɺ͕࣍੒ཱ͢Δɻ
    |S| = [M : F] = |Aut(M/F)| (34)
    (33) (34) ΑΓɺ
    S = Aut(M/F)
    ͕੒ཱ͢Δɻ
    ɾ(c) ⇒ (b)
    (c) ͸ɺ೚ҙͷ σ ∈ G ʹ͍ͭͯɺఆٛҬΛ M ʹ੍ݶͨ͠΋ͷ͕ɺM ͷࣗݾಉܕࣸ૾ɺ͢ͳΘͪɺશ୯ࣹ
    ͳࣸ૾Ͱ͋Δ͜ͱΛ͓ࣔͯ͠Γɺैͬͯɺσ(M) = M ͕੒ཱ͢Δɻ
    25

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  26. ɾ(b) ⇒ (c)
    Aut(M/F) ʹଐ͢Δࣸ૾͸ɺ஋Ҭͱͯ͠ͷ M Λ E ʹຒΊࠐΜͰߟ͑ΔͱɺHomF
    (M, E) ʹଐ͢Δࣸ૾
    Ͱ΋͋Γɺ
    Aut(M/F) ⊆ HomF
    (M, E)
    ͱ͍͏แؚؔ܎͕੒ཱ͢Δɻैͬͯɺิ୊ 4 ͱ͋Θͤͯɺ
    S = HomF
    (M, E) ⊇ Aut(M/F) (35)
    ͕ಘΒΕΔɻҰํɺ(b) ͸ɺσ ∈ G ͷఆٛҬΛ M ʹ੍ݶͨ͠΋ͷ͕ M ͷࣗݾಉܕࣸ૾Ͱ͋Δ͜ͱΛࣔͯ͠
    ͓Γɺ͞Βʹ F ͷݩΛಈ͔͞ͳ͍ͱ͍͏৚͔݅Βɺ೚ҙͷ σ ∈ G ʹ͍ͭͯɺ
    σ|M
    ∈ Aut(M/F)
    ͕੒ཱ͢Δɻ͜Ε͸ɺ
    S ⊆ Aut(M/F) (36)
    Λද͓ͯ͠Γɺ(35) (36) ΑΓɺ
    S = Aut(M/F)
    ͕ಘΒΕΔɻ
    ɾ(b) ⇒ (a)
    (b) ͕੒ཱ͢Δ࣌ɺิ୊ 5 ΑΓɺ
    ∀σ ∈ G; Aut(E/M) = σAut(E/M)σ−1
    ͕੒ΓཱͭͷͰɺAut(E/M) ͸ G ͷਖ਼ن෦෼܈ͱͳΔɻ
    ɾ(a) ⇒ (b)
    ೚ҙͷ σ ∈ G ʹ͍ͭͯɺ͜Ε͸ E ্ͷࣗݾಉܕࣸ૾Ͱ͋ΓɺF ͷݩΛಈ͔͞ͳ͍͜ͱ͔Βɺ
    E ⊃ σ(M) ⊃ F
    ͕੒Γཱͪɺఆཧ 8 ΑΓ E/σ(M) ͸ΨϩΞ֦େͰ͋Γɺ
    σ(M) = EAut(E/σ(M)) (37)
    ͕੒ཱ͢Δɻಉ༷ʹͯ͠ɺE/M ΋ΨϩΞ֦େͰ͋Γɺ
    M = EAut(E/M) (38)
    ͕੒ཱ͢Δɻ͜͜ͰɺH = Aut(E/M) ͕ G ͷਖ਼ن෦෼܈ͩͱ͢Δͱɺ
    ∀σ ∈ G; Aut(E/M) = σAut(E/M)σ−1
    ͕੒ΓཱͭͷͰɺิ୊ 5 ͱ͋Θͤͯɺ
    ∀σ ∈ G; Aut (E/σ(M)) = Aut(E/M)
    ͕ಘΒΕΔɻ͜ͷؔ܎Λ (37) (38) ʹద༻͢Δͱɺ
    ∀σ ∈ G; σ(M) = M
    ͕ಘΒΕΔɻ ˙
    26

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  27. 5 Մղ܈ͷੑ࣭
    5.1 ܈ͷಉܕఆཧ
    ͜ͷޙɺՄղ܈ʹؔ͢ΔఆཧΛূ໌͢Δࡍʹɺ܈ͷୈҰಉܕఆཧɺ͓ΑͼɺୈࡾಉܕఆཧΛ༻͍·͢ɻ೦ͷ
    ͨΊɺ͜ΕΒͷఆཧΛ͜͜ʹهࡌ͓͖ͯ͠·͢ɻ
    ୈҰಉܕఆཧ
    G ͱ H Λ܈ͱͯ͠ɺf : G −→ H Λ܈ͷ४ಉܕࣸ૾ͱ͢Δɻ͜ͷ࣌ɺ࣍ͷؔ܎͕੒ཱ͢Δɻ
    • Ker f ͸ G ͷਖ਼ن෦෼܈Ͱ͋Δɻ
    • Im f ͸ H ͷ෦෼܈Ͱ͋Δɻ
    • ܈ͷಉܕ G/Ker ∼
    = Im f ͕੒ཱ͢Δɻ
    ୈࡾಉܕఆཧ
    G Λ܈ͱ͢ΔɻG′ ͱ N ͸ͲͪΒ΋ G ͷਖ਼ن෦෼܈Ͱɺแؚؔ܎ G ⊇ G′ ⊇ N ͕੒ཱ͢Δͱ͢Δɻ͜ͷ
    ࣌ɺ࣍ͷؔ܎͕੒ཱ͢Δɻ
    • ৒༨܈ G′/N ͸৒༨܈ G/N ͷਖ਼ن෦෼܈Ͱ͋Δɻ
    • ܈ͷಉܕ (G/N)/(G′/N) ∼
    = G/G′ ͕੒ཱ͢Δɻ
    5.2 Մղ܈ͷఆٛ
    લઅͷٞ࿦ʹΑΓɺM/F ͕ΨϩΞ֦େͰ͋Δ͜ͱͱɺH = Aut(E/M) ͕ G = Aut(E/F) ͷਖ਼ن෦෼܈
    ʹͳΔ͜ͱ͕ಉ஋Ͱ͋Δͱ෼͔Γ·ͨ͠ɻͦͯ͠ɺલड़ͷΑ͏ʹɺH ͕ G ͷਖ਼ن෦෼܈Ͱ͋Ε͹ɺ৒༨܈
    G/H ͕ಘΒΕ·͢ɻ͜ͷ࣌ɺG ͔Βଞͷ܈ʹର͢Δ४ಉܕࣸ૾ f ͰɺKer f = H ͱͳΔ΋ͷ͕͋Ε͹ɺ܈
    ͷୈҰಉܕఆཧʹΑΓɺIm f ͸৒༨܈ G/H ͱಉܕʹͳΓ·͢ɻ͜ΕΛར༻͢Δͱɺ࣍ͷఆཧ͕ಘΒΕ·͢ɻ
    ఆཧ 11 ΨϩΞ֦େ E/F ͷதؒମ M ʹ͓͍ͯɺM/F ͕ΨϩΞ֦େͰ͋Δ࣌ɺ
    G = Aut(E/F)
    H = Aut(E/M)
    ͱͯ͠ɺ࣍ͷ܈ಉܕ͕੒ཱ͢Δɻ
    G/H ∼
    = Aut(M/F)
    ʢূ໌ʣ
    M/F ͕ΨϩΞ֦େͰ͋Δ͜ͱ͔Βɺఆཧ 10 ͷ (c) ͷ৚͕݅੒ΓཱͭͷͰɺ࣍ͷࣸ૾ f ͕ఆٛͰ͖ͯɺ
    f : G −→ Aut(M/F)
    σ −→ σ|M
    ͜Ε͸શࣹʹͳΔͷͰɺ
    Im f = Aut(M/F) (39)
    27

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  28. ͕੒ཱ͢Δɻ͞Βʹɺఆ͔ٛΒ໌Β͔ͳΑ͏ʹɺf ͸܈ͷؒͷ४ಉܕࣸ૾Λ༩͑Δɻ·ͨɺAut(M/F) ͷ܈
    ͱͯ͠ͷ୯Ґݩ͸ M ্ͷ߃౳ࣸ૾ σ|M
    = idM
    Ͱ͋Γɺ͜ͷݪ૾͸ σ ∈ H = Aut(E/M) ʹΑͬͯ༩͑ΒΕ
    Δɻ͢ͳΘͪɺ
    Ker f = H (40)
    ͕੒Γཱͭɻैͬͯɺ܈ͷୈҰಉܕఆཧΑΓɺ
    G/Ker f ∼
    = Im f
    ͕੒ཱ͢ΔͷͰɺ͜Εʹ (39) ͱ (40) Λ୅ೖͯ͠ɺ
    G/H ∼
    = Aut(M/F)
    ͕ಘΒΕΔɻ ˙
    ఆཧ 11 ͸ɺM/F ͕ΨϩΞ֦େͰ͋Δ࣌ɺ͢ͳΘͪɺE/F ͱ E/M ͷ 2 ͭͷ֦େʹର͢ΔΨϩΞ܈ G ͱ
    H ʹ͓͍ͯɺH ͕ G ͷਖ਼ن෦෼܈ʹͳ͍ͬͯΔ࣌ɺ֦େ M/F ͷΨϩΞ܈ͷߏ଄͕৒༨܈ G/H ͱܾͯ͠
    ·Δ͜ͱΛҙຯ͠·͢ɻͦͯ͠ɺ·ͩগ͠ઌʹͳΓ·͕͢ɺ
    ʮ7.2 ୅਺తʹՄղͳଟ߲ࣜʯʹ͓͍ͯɺ୅਺ํఔ
    ࣜͷՄղੑΛٞ࿦͢Δࡍ͸ɺΨϩΞ֦େ E/F ʹର͢Δதؒମͷྻ
    E = A0
    ⊃ A1
    ⊃ · · · ⊃ Al
    = F
    ͱɺͦΕʹ൐͏ΨϩΞ܈ G = Aut(E/F) ͷ෦෼܈ͷྻ
    G = Aut(E/Al
    ) ⊃ Aut(E/Al−1
    ) ⊃ · · · ⊃ Aut(E/A0
    ) = {1}
    ͕ొ৔͠·͢ɻ͜ͷ࣌ɺఆཧ 8 ΑΓɺ೚ҙͷதؒମ Ai
    ʹର֦ͯ͠େ E/Ai
    ͸ΨϩΞ֦େͱͳΓ·͢ɻͦ͜
    Ͱɺ೚ҙͷྡΓ߹͏ϖΞ Ai
    ͱ Ai+1
    ΛऔΓग़ͯ͠ɺ
    E ⊃ Ai
    ⊃ Ai+1
    ͱ͍͏૊Έ߹ΘͤΛߟ͑Δͱɺ֦େ E/Ai+1
    ͷΨϩΞ܈ Gi+1
    = Aut(E/Ai+1
    ) ʹରͯ͠ɺ֦େ E/Ai
    ͷΨϩ
    Ξ܈ Gi
    = Aut(E/Ai
    ) ͕ਖ਼ن෦෼܈ʹͳ͍ͬͯΕ͹ɺ֦େ Ai
    /Ai+1
    ΋ΨϩΞ֦େͰ͋Γɺ
    Aut(Ai
    /Ai+1
    ) ∼
    = Gi+1
    /Gi
    ͱ͍͏܈ͷಉܕ͕੒Γཱͪ·͢ɻͦͯ͠ಛʹɺ͢΂ͯͷ Aut(Ai
    /Ai+1
    ) ͕Ξʔϕϧ܈ʢੵ͕Մ׵ͳ܈ʣʹͳͬ
    ͍ͯΔ৔߹͕ॏཁͳҙຯΛ࣋ͭ͜ͱʹͳΓ·͢ɻ͜ΕΛ೦಄ʹ͓͍ͯɺՄղ܈Λ࣍ͷΑ͏ʹఆ͓͖ٛͯ͠
    ·͢ɻ
    ఆٛ 1 ܈ G ͕࣍ͷ 3 ͭͷ৚݅Λຬͨ࣌͢ɺ͜ΕΛՄղ܈ͱݺͿɻ
    1. ༗ݶݸͷ෦෼܈ͷྻ G = Gl
    ⊃ Gl−1
    ⊃ · · · ⊃ G0
    = {1} Λ࣋ͭɻ
    2. ྡΓ߹͏෦෼܈͸͢΂ͯਖ਼ن෦෼܈Ͱ͋Δɻ͢ͳΘͪɺi = 0, · · · , l − 1 ʹରͯ͠ɺGi
    ͸ Gi+1
    ͷਖ਼ن
    ෦෼܈Ͱ͋Δɻ
    3. ৒༨܈ Gl
    /Gl−1
    , Gl−1
    /Gl−2
    , · · · , G1
    /G0
    ͸͢΂ͯΞʔϕϧ܈Ͱ͋Δɻ
    Մղ܈ͷఆٛʹ͓͍ͯɺG ʹର͢Δ෦෼܈ͷྻͷऔΓํ͸ࣄલʹࢦఆ͞Ε͍ͯΔΘ͚Ͱ͸͋Γ·ͤΜɻԿ
    ͔ 1 ͭͰ΋ 2. ͱ 3. ͷ৚݅Λຬͨ͢༗ݶͳ෦෼܈ͷྻ͕ଘࡏ͢Ε͹ɺG ͸Մղ܈Ͱ͋Δ͜ͱʹͳΓ·͢ɻͭ·
    ΓɺՄղ܈Ͱ͋Δ͔Ͳ͏͔͸ɺ܈ G ͦͷ΋ͷ͕͍࣋ͬͯΔੑ࣭ʹͳΓ·͢ɻ࣍અͰ͸ɺՄղ܈ G ʹ͍ͭͯɺ
    ͦͷఆ͔ٛΒҰൠʹ੒ΓཱͭఆཧΛࣔ͠·͢ɻ
    28

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  29. 5.3 Մղ܈ͷҰൠతੑ࣭
    Մղ܈ G ͕ਖ਼ن෦෼܈ N Λ࣋ͭ৔߹ɺࣗવͳࣹӨ f : G −→ G/N Λ௨ͯ͠ɺ৒༨܈ G/N ʹ΋෦෼܈ͷ
    ྻΛಋೖ͢Δ͜ͱ͕Ͱ͖ɺG/N ΋·ͨՄղ܈ʹͳΓ·͢ɻ͜ΕΛࣔ͢ͷ͕࣍ͷఆཧͰ͢ɻ
    ఆཧ 12 ܈ G ͕ਖ਼ن෦෼܈ N Λ࣋ͭ࣌ɺG ͕Մղ܈Ͱ͋Ε͹ɺ৒༨܈ G/N ΋Մղ܈ʹͳΔɻ
    ʢূ໌ʣ
    G ͔Β G/N ΁ͷࣗવͳࣹӨΛ f ͱ͢ΔɻG ͕Մղ܈Ͱ͋Δ͜ͱ͔Βɺਖ਼ن෦෼܈ͷྻ
    G = Gl
    ⊃ Gl−1
    ⊃ · · · ⊃ G0
    = {1}
    ͕ଘࡏͯ͠ɺ
    ྡΓ߹͏܈ͷ৒༨܈͕Ξʔϕϧ܈ͱͳΔɻ͜ͷ࣌ɺ
    f ͕४ಉܕࣸ૾Ͱ͋Δ͜ͱ͔Βɺ
    i = 0, · · · , l−1
    ʹରͯ͠ɺf(Gi
    ) ͸ f(Gi+1
    ) ͷਖ਼ن෦෼܈ͱͳΔ͜ͱ͕༰қʹ֬ೝͰ͖ͯɺਖ਼ن෦෼܈ͷྻ
    G/N = f(Gl
    ) ⊃ f(Gl−1
    ) ⊃ · · · ⊃ f(G0
    ) = {1}
    ͕ߏ੒Ͱ͖Δɻ͞ΒʹɺྡΓ߹͏෦෼܈ͷ৒༨܈ʹରͯ͠ɺ࣍ͷ४ಉܕࣸ૾ f ͕ఆٛͰ͖Δɻ
    f : Gi+1
    /Gi
    −→ f(Gi+1
    )/f(Gi
    )
    gGi
    −→ f(g)f(Gi
    )
    f ͕શࣹͰ͋Δ͜ͱ͔Βɺf ΋શࣹͰ͋ΓɺGi+1
    /Gi
    ͕Ξʔϕϧ܈Ͱ͋Δ͜ͱ͔Βɺ४ಉܕࣸ૾ f Λ௨ͯ͠ɺ
    f(Gi+1
    )/f(Gi
    ) ΋Ξʔϕϧ܈ͱͳΔ͜ͱ͕෼͔Δɻ ˙
    ఆཧ 12 ͸ɺG ͕Մղ܈Ͱ͋Ε͹ G/N ΋Մղ܈ʹͳΔͱ͍͏΋ͷͰͨ͠ɻͦͯ͠ɺN ͕Մղ܈Ͱ͋Ε͹ɺ
    ͦͷٯɺ͢ͳΘͪɺG/N ͕Մղ܈Ͱ͋Ε͹ G ΋Մղ܈ʹͳΔ͜ͱ͕࣍ͷఆཧͰࣔ͞Ε·͢ɻ
    ఆཧ 13 ܈ G ͕ਖ਼ن෦෼܈ N Λ࣋ͭ࣌ɺN ͱ G/N ͕ͲͪΒ΋Մղ܈Ͱ͋Ε͹ɺG ΋Մղ܈ʹͳΔɻ
    ʢূ໌ʣ
    ४උͱͯ͠ɺҰൠʹɺ৒༨܈ G/N ͷ೚ҙͷ෦෼܈ H ʹରͯ͠ɺ
    H ∼
    = G′/N (41)
    Λຬͨ͢෦෼܈ G′ ͕ߏ੒Ͱ͖ͯɺ͞ΒʹɺH ͕ G/N ͷਖ਼ن෦෼܈Ͱ͋Ε͹ɺG′ ͸ G ͷਖ਼ن෦෼܈Ͱɺ
    (G/N)/(G′/N) ∼
    = G/G′ (42)
    ͕੒Γཱͭ͜ͱΛࣔ͢ɻ·ͣɺG ͔Β G/N ΁ͷࣗવͳࣹӨ f Λ༻͍ͯɺ
    G′ = {g ∈ G | f(g) ∈ H}
    ͱఆٛ͢ΔͱɺN ͷཁૉ͸ f ʹΑͬͯ H ͷ୯ҐݩʹҠΔ͜ͱ͔Βɺ
    G ⊇ G′ ⊇ N
    Ͱ͋Γɺf ͷ४ಉܕੑΛར༻ͯ͠ɺG′ ͸ G ͷ෦෼܈ʹͳΔ͜ͱ͕༰қʹ෼͔Δɻ͞Βʹɺf ͷఆٛҬΛ G′
    ʹ੍ݶͨ͠΋ͷΛ f|G′
    ͱ͢Δͱɺ
    Ker f|G′
    = N, Im f|G′
    = H
    29

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  30. ͱͳΔ͜ͱ΋ఆ͔ٛΒ༰қʹ෼͔Δɻैͬͯɺ܈ͷୈҰಉܕఆཧʹΑΓ (41) ͕੒ཱ͢Δɻ࣍ʹɺH ͕ਖ਼ن෦
    ෼܈ͩͱ͢Δͱɺ೚ҙͷ g ∈ G ͓Αͼ g′ ∈ G′ ʹରͯ͠ɺ
    f(gg′g−1) = f(g)f(g′)f(g−1) ∈ H
    ͕੒ཱ͢Δɻ
    ʢ͜͜Ͱ͸ɺf(g), f(g−1) ∈ G/Nɺf(g′) ∈ H Ͱ͋ΓɺH ͸ G/N ͷਖ਼ن෦෼܈Ͱ͋Δͱ͍͏
    ࣄ࣮Λ༻͍ͨɻ
    ʣ͜ΕΑΓɺ
    gg′g−1 ∈ G′
    ͕੒ΓཱͭͷͰɺG′ ͸ G ͷਖ਼ن෦෼܈Ͱ΋͋Δɻैͬͯɺ܈ͷୈࡾಉܕఆཧΑΓɺG′/N ͸ G/N ͷਖ਼ن෦
    ෼܈Ͱ͋Γɺ(42) ͕੒ཱ͢Δɻ͜ΕͰ४උ͕Ͱ͖ͨɻ
    ࠓɺG/N ͕Մղ܈Ͱ͋Δ͜ͱ͔Βɺ෦෼܈ͷྻ
    G/N = Hl
    ⊃ Hl−1
    ⊃ · · · ⊃ H0
    = {1}
    ͕ଘࡏ͢Δ͕ɺͦΕͧΕͷ Hi
    ʹରͯ͠ɺલड़ͷٞ࿦Λద༻͢ΔͱɺHi

    = Gi
    /N ͱͳΔ G ͷ෦෼܈ Gi
    ͕ଘ
    ࡏͯ͠ɺ
    G/N = Gl
    /N ⊃ Gl−1
    /N ⊃ · · · ⊃ G0
    /N = N/N = {1}
    ͱॻ͖ද͢͜ͱ͕Ͱ͖Δɻ͜͜ͰɺྡΓ߹͏෦෼܈͸ਖ਼ن෦෼܈Ͱɺ͔ͭɺ৒༨܈ (Gi+1
    /N)/(Gi
    /N) ͸Ξʔ
    ϕϧ܈ͱͳΔɻ͜͜Ͱɺ৒༨܈ Gi+1
    /N ͱͦͷਖ਼ن෦෼܈ H = Gi
    /N ʹରͯ͠લड़ͷٞ࿦Λద༻͢Δͱɺ
    H = Gi
    /N ∼
    = G′/N (43)
    ͱͳΔ Gi+1
    ͷਖ਼ن෦෼܈ G′ ͕ଘࡏͯ͠ɺ
    (Gi+1
    /N)/(G′/N) ∼
    = Gi+1
    /G′
    ͕੒ཱ͢Δɻ͜͜Ͱɺ(43) Λຬͨ͢ G ͷ෦෼܈ G′ ͸໌Β͔ʹ G′ = Gi
    ͳͷͰɺ
    (Gi+1
    /N)/(Gi
    /N) ∼
    = Gi+1
    /Gi
    ͱͳΓɺ্ࣜͷࠨล͕Ξʔϕϧ܈Ͱ͋Δ͜ͱ͔Βɺӈล΋Ξʔϕϧ܈ͱͳΔɻͭ·ΓɺG ʹରͯ͠ɺਖ਼ن෦෼
    ܈ͷྻ
    G = Gl
    ⊃ Gl−1
    ⊃ · · · ⊃ G0
    = N (44)
    ͕ଘࡏͯ͠ɺྡΓ߹͏෦෼܈ Gi+1
    ͱ Gi
    ͷ৒༨܈ Gi+1
    /Gi
    ͸Ξʔϕϧ܈ͱͳΔ͜ͱ͕ࣔ͞Εͨɻ࠷ޙʹɺ
    N ͕Մղ܈Ͱ͋Δ͜ͱ͔Βɺಉ༷ͷਖ਼ن෦෼܈ͷྻ
    N = Nr
    ⊃ Nr−1
    ⊃ · · · ⊃ {1} (45)
    ͕ଘࡏͯ͠ɺ(44) ͱ (45) Λͭͳ͛Δ͜ͱͰɺG ͸Մղ܈ͷ৚݅Λຬͨ͢͜ͱʹͳΔɻ ˙
    6 ଟ߲ࣜͷ࠷খ෼ղମ
    6.1 ଟ߲ࣜͷࠜʹΑΔ֦େ
    ͜Ε·Ͱɺମͷ֦େ E/F ʹͱ΋ͳ͏ΨϩΞ܈ͷߏ଄Λஸೡʹௐ΂͖ͯ·ͨ͠ɻ͔͜͜Β͸ɺ͍Α͍Αଟ߲
    ࣜͷߏ଄ͱମͷ֦େͷؔ܎Λ໌Β͔ʹ͍͖ͯ͠·͢ɻ
    ʮ3.1 ମͷࣗݾಉܕ܈ͱΨϩΞ܈ʯͰ͸ɺ༗ཧ਺ମ Q ʹ
    30

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  31. ํఔࣜͷղΛ෇͚Ճ֦͑ͯେ͢Δͱ͍͏ߟ͑ํΛ঺հ͠·͕ͨ͠ɺ͜͜Ͱ͸ɺҰൠʹɺ೚ҙͷଟ߲ࣜ f(x) ʹ
    ରͯ͠ɺf(x) = 0 ͷղΛ͢΂ͯ෇͚Ճ֦͑ͨେମ͕ߏ੒Ͱ͖Δ͜ͱΛࣔ͠·͢ɻ
    ͦͷ४උͱͯ͠ɺ·ͣ͸ɺఆཧ 2 Ͱ༩֦͑ͨେମ F(α) ͷఆٛΛਖ਼֬ʹݟ௚͓͖ͯ͠·͢ɻҰൠʹɺF ͷ֦
    େମ E ʹ͓͍ͯɺα ∈ E Λ୅਺తͳݩͱͯ͠ɺ࠷খଟ߲ࣜ Irr(α, F) ͷ࣍਺Λ n ͱ͢Δ࣌ɺू߹
    F(α) = {a0
    + a1
    α + · · · + an−1
    αn−1 | a0
    , · · · , an−1
    ∈ F}
    Λ F ͷ֦େମͱݟͳ͢͜ͱ͕Ͱ͖·ͨ͠ɻ͜ͷ࣌ɺF(α) ͷݩͷੵΛܭࢉ͢Δࡍ͸ɺp(x) Λ࠷খଟ߲ࣜ
    Irr(α, F) ͱͯ͠ɺp(α) = 0 ͷ৚͔݅Βɺn ࣍Ҏ্ͷ߲͸ɺn − 1 ࣍ҎԼʹॻ͖௚͢ͱ͍͏৚͕݅͋Γ·ͨ͠ɻ
    ͜Ε͸ɺม਺ x ͷʢ࣍਺Λ੍ݶ͠ͳ͍ʣଟ߲ࣜશମͷू߹ F[x] ʹରͯ͠ɺp(x) Ͱׂͬͨ༨Γ r(x) ͕౳͍͠
    ΋ͷΛಉҰࢹ͢Δͱ͍͏ಉ஋ྨΛೖΕͨ৒༨ମ F[x]/p(x) ͱಉܕʹͳΔ͜ͱ͕෼͔Γ·͢ɻ۩ମతʹ͸ɺ࣍
    ͷࣸ૾͕ಉܕࣸ૾Λ༩͑·͢ɻ
    F[x]/p(x) −→ F(α)
    f(x) −→ r(α) (46)
    ͜͜ʹɺf(x) ͸ ৒༨߲Λ r(x) ͱ͢Δ F[x]/p(x) ͷಉ஋ྨͷ 1 ͭɺ͢ͳΘͪɺ
    f(x) = g(x)p(x) + r(x) ʢr(x) ͸ n − 1 ࣍ҎԼͷଟ߲ࣜʣ
    ͱද͞ΕΔଟ߲ࣜͱ͠·͢ɻ
    ͦͯ͠ɺ࣮͸ɺ͜ͷؔ܎ͦͷ΋ͷΛ F(α) ͷఆٛͱΈͳ͢͜ͱ͕Ͱ͖·͢ɻ͜Ε·Ͱ F(α) Λఆٛ͢Δࡍ
    ͸ɺα ΛؚΉ F ͷ֦େମ E ͷଘࡏΛલఏͱ͍ͯ͠·͕ͨ͠ɺ࣍ͷखଓ͖Λ౿Ί͹ɺE ͷଘࡏΛԾఆ͢Δ͜ͱ
    ͳ͘ɺF(α) Λఆٛ͢Δ͜ͱ͕ՄೳʹͳΓ·͢ɻ
    ·ͣ͸͡ΊʹɺF ্ͷط໿ଟ߲ࣜ p(x) Λ༻ҙͯ͠ɺଟ߲ࣜͷ৒༨ମ F[x]/p(x) Λఆٛ͠·͢ɻ೦ͷͨΊ
    ʹɺੵͷٯݩͷଘࡏΛ֬ೝ͓ͯ͘͠ͱɺ࣍ͷΑ͏ʹͳΓ·͢ɻҰൠʹɺ͋Δݩͷ৒༨߲Λ r(x) ͱ͢Δͱɺr(x)
    ͷ࣍਺͸ p(x) ͷ࣍਺ΑΓখ͘͞ɺ͔ͭɺp(x) ͸ن໿ଟ߲ࣜͳͷͰɺr(x) ͱ p(x) ͷ࠷େެ໿ࣜ͸ 1 ͱͳΓɺ
    ϢʔΫϦουͷޓআ๏ΑΓɺ࣍Λຬͨ͢ଟ߲ࣜ a(x), b(x) ∈ F[x] ͕ଘࡏ͠·͢ɻ
    r(x)a(x) + p(x)b(x) = 1
    ͜Ε͸ɺr(x)a(x) Λ p(x) Ͱׂͬͨ࣌ͷ৒༨߲͕ 1 ʹͳΔ͜ͱΛ͓ࣔͯ͠Γɺa(x) ͷಉ஋ྨ͕ r(x) ͷಉ஋
    ྨʹର͢ΔੵͷٯݩͱͳΓ·͢ɻଓ͍ͯɺه߸ α Λ༻͍ͨܗࣜతͳଟ߲ࣜͷू߹
    F(α) = {a0
    + a1
    α + · · · + an−1
    αn−1 | a0
    , · · · , an−1
    ∈ F}
    Λఆ͓͖ٛͯ͠ɺ(46) ͷࣸ૾Λ௨ͯ͠ɺଟ߲ࣜ f(x) ͷ৒༨߲ r(x) Λه߸ α ͷଟ߲ࣜ r(α) ͱಉҰࢹ͢Δ͜
    ͱͰɺF(α) Λ F[x]/p(x) ͱಉܕͳମͱݟͳ͠·͢ɻ͜ΕͰɺ֦େମ E ͷଘࡏΛԾఆ͢Δ͜ͱͳ͘ɺ
    F(α) ∼
    = F[x]/p(x)
    Λఆٛ͢Δ͜ͱ͕Ͱ͖·ͨ͠ɻ͜ͷఆٛʹ͓͍ͯ͸ɺα ͦͷ΋ͷͰ͸ͳ͘ɺͦΕʹ෇ਵ͢Δن໿ଟ߲ࣜ p(x)
    ͕ຊ࣭తͳ໾ׂΛՌ͍ͨͯ͠Δ͜ͱ͕෼͔Γ·͢*8ɻ͜ͷΑ͏ʹͯ͠ɺF(α) ͕ఆٛͰ͖Ε͹ɺ͞Βʹผͷ
    F ্ͷن໿ଟ߲ࣜ p′(x) ʹ͍ͭͯɺ͜ΕΛ F(α) ্ͷط໿ଟ߲ࣜͱΈͳͯ͠ಉٞ͡࿦Λ܁Γฦ͢͜ͱͰɺ
    F(α)[x]/p′(x) ͱಉܕͳମͱͯ͠ F(α, α′) Λఆٛ͢Δ͜ͱ΋ՄೳʹͳΓ·͢*9ɻ
    *8 ͦͷҙຯͰ͸ɺF(α) Ͱ͸ͳ͘ɺF(p(x)) ͱͰ΋දه͢Δ΂͖Ͱ͕͢ɺ͜͜Ͱ͸ɺα ʹ൐͏ط໿ଟ߲ࣜ p(x) ͷଘࡏ͕҉໧ʹ૝ఆ
    ͞Ε͍ͯΔ΋ͷͱ͍ͯͩ͘͠͞ɻ
    *9 ֦େ͢Δॱ൪ʹΑͬͯ݁Ռ͕มΘΒͳ͍ࣄ͸ɺ
    ʮ6.3 ࠷খ෼ղମͷҰҙੑʯͰࣔ͞Ε·͢ɻ
    31

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  32. ·ͨɺ
    ͜Ε·Ͱɺ
    ʮf(x) = 0 ͷղ α Λ෇͚Ճ֦͑ͨେମʯ
    ͱ͍͏දݱΛ͖ͯ͠·ͨ͠ɻ͜Ε΋·ͨɺ
    f(x) = 0
    ͷղΛؚΉ֦େମ E ͷଘࡏΛલఏͱͨ͠ߟ͑ํʹͳΓ·͕͢ɺଟ߲ࣜͷҼ਺෼ղΛར༻͢Δͱɺ֦େମ E ͷ
    ଘࡏΛԾఆͤͣʹɺ͜ΕΛදݱ͢Δ͜ͱ͕Ͱ͖·͢ɻͨͱ͑͹ɺଟ߲ࣜ
    f(x) = x2 − 2
    ͸ɺ༗ཧ਺ମ Q Λ܎਺ͱ͢Δଟ߲ࣜͱݟͳ͢ݶΓɺ͜ΕҎ্ɺҼ਺෼ղ͢Δ͜ͱ͸Ͱ͖·ͤΜɻ͔͠͠ͳ͕
    Βɺن໿ଟ߲ࣜͰ͋Δ f(x) Λ༻͍ͯߏ੒֦ͨ͠େମ
    Q(α) ∼
    = Q[x]/f(x)
    Λ܎਺ͱ͢Δଟ߲ࣜͱΈͳͤ͹ɺ
    f(x) = (x − α)(x + α)
    ͱҼ਺෼ղ͢Δ͜ͱ͕Ͱ͖·͢ɻ͜ͷखଓʹ͓͍ͯɺα ͸ɺຊ࣭తʹ͸ɺ࣮਺ମͷݩͰ͋Δ

    2 ͱಉ͡໾ׂΛ
    ͍ͯ͠·͕͢ɺ͜ͷٞ࿦ͦͷ΋ͷ͸࣮਺ମ R ͷଘࡏΛԾఆͤͣͱ΋੒Γཱͭ఺ʹ஫ҙ͍ͯͩ͘͠͞ɻ͜ͷΑ
    ͏ͳҙຯʹ͓͍ͯɺଟ߲ࣜΛҼ਺෼ղͨ͠ࡍʹಘΒΕΔ֦େମͷݩ α Λଟ߲ࣜͷࠜͱݺͼ·͢ɻ
    ͦͯ͠ɺಉ͘͡ɺ͜ͷΑ͏ͳҙຯʹ͓͍ͯɺ೚ҙͷଟ߲ࣜ f(x) ʹ͍ͭͯɺͦΕΛ׬શʹҼ਺෼ղ͢ΔΑ͏
    ͳ֦େମ͕ߏ੒Ͱ͖Δͱ͍͏ͷ͕ɺ࣍ͷఆཧͷओுʹͳΓ·͢ɻ
    ఆཧ 14 ମ F ্ͷ೚ҙͷଟ߲ࣜ f(x) ∈ F[x] ʹରͯ͠ɺ͜ΕΛ࣍ͷΑ͏ʹҼ਺෼ղՄೳʹ͢Δ֦େମ
    F(α1
    , · · · , αn
    ) Λߏ੒͢Δ͜ͱ͕Ͱ͖Δɻ
    f(x) = a(x − α1
    )(x − α2
    ) · · · (x − αn
    ) (a ∈ F)
    ʢূ໌ʣ
    f(x) ͕ F ্Ͱن໿Ͱͳ͍৔߹͸ɺf(x) = g(x)h(x) · · · ͱط໿ଟ߲ࣜʹ෼ղͯ͠ɺͦΕͧΕʹ͍ͭͯఆཧ͕
    ূ໌Ͱ͖Ε͹े෼Ͱ͋Δɻ
    ʢͦΕͧΕͷن໿ଟ߲ࣜʹ͍ͭͯಘΒΕͨࠜΛ͢΂ͯ෇Ճ֦ͨ͠େମΛߏ੒͢Δɻ
    ʣ
    f(x) ͕ن໿ͳ৔߹ɺf(x) ͷ࠷ߴ࣍਺ͷ܎਺Λ 1 ͱͨ͠΋ͷΛ p(x) ͱͯ͠ɺ֦େମ
    F(α1
    ) ∼
    = F[x]/p(x)
    Λߏ੒͢Δɻ͜Ε͸ɺه߸ α1
    ͷଟ߲ࣜ r(α1
    ) Λ৒༨ྨΛ r(x) ͱ͢Δ৒༨ମ F[x]/p(x) ͷݩͱಉҰࢹͯ͠ಘ
    ΒΕΔମͰ͋Δɻͦͯ͠ɺ͜ͷମͷԼͰ͸ɺf(α1
    ) = 0 ͕੒ཱ͢ΔͷͰɺଟ߲ࣜͷ৒༨ఆཧʹΑΓɺ
    f(x) = a(x − α1
    )g(x) (a ∈ F, g(x) ∈ F(α1
    )[x])
    ͱɺf(x) ΛҼ਺෼ղ͢Δଟ߲ࣜ g(x) ͕ଘࡏ͢Δɻ͜͜Ͱɺg(x) ͷ࠷ߴ࣍਺ͷ܎਺͕ 1 ʹͳΔΑ͏ʹɺa Λ
    બͿ΋ͷͱ͢Δɻ
    ଓ͍ͯɺg(x) ∈ F(α1
    )[x] ʹಉ༷ͷٞ࿦Λద༻͢Δͱɺ৽ͨͳମ F(α1
    , α2
    ) Λߏ੒ͯ͠ɺ
    g(x) = (x − α2
    )h(x) (h(x) ∈ F(α1
    , α2
    )[x])
    ͱҼ਺෼ղͰ͖ΔɻҎԼɺಉ༷ͷٞ࿦Λ܁Γฦͤ͹Α͍ɻ ˙
    ఆཧ 14 ͷखଓ͖ʹΑͬͯߏ੒͞ΕΔ֦େମ F(α1
    , · · · , αn
    ) Λଟ߲ࣜ f(x) ͷ࠷খ෼ղମͱݺͼ·͢ɻҰൠ
    ʹɺଟ߲ࣜ f(x) Λ 1 ࣍ࣜͷੵʹ෼ղՄೳʹ͢Δ೚ҙͷ֦େମΛ f(x) ͷ෼ղମͱ͍͍·͕͢ɺͦͷதͰ΋࠷
    32

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  33. খͷ΋ͷͱ͍͏ҙຯʹͳΓ·͢ɻ
    ྫ 6-1

    ༗ཧ਺ମ Q ্ͷن໿ଟ߲ࣜ
    f(x) = x3 − 2
    ʹରͯ͠ɺఆཧ 14 ͷূ໌Ͱ༩͑ͨखଓ͖Λ࣮ࡍʹద༻ͯ͠Έ·͢ɻ·ͣɺܗࣜతʹه߸ α ΛՃ͑ͨମ
    Q(α) Λ৒༨ମ Q[x]/f(x) ͱಉҰࢹ͢Δ͜ͱͰఆٛ͠·͢ɻ
    Q(α) ∼
    = Q[x]/f(x)
    ͜ͷ࣌ɺf(α) = 0 ͱ͍͏ϧʔϧΛ༻͍Δͱɺମ Q(α) ͷ্Ͱɺf(x) ͸࣍ͷΑ͏ʹҼ਺෼ղ͞ΕΔ͜ͱ
    ͕ɺ௚઀ͷܭࢉͰ֬ೝͰ͖·͢ɻ
    f(x) = (x − α)g(x)
    g(x) = x2 + αx + α2
    ͦ͜Ͱ͞ΒʹɺQ(α) ʹରͯ͠ɺܗࣜతʹه߸ β ΛՃ͑ͨମ Q(α, β) Λ৒༨ମ Q(α)[x]/g(x) ͱಉҰࢹ
    ͢Δ͜ͱͰఆٛ͠·͢ɻ
    Q(α, β) ∼
    = Q(α)[x]/g(x)
    ͜ͷ࣌ɺg(β) = 0 ͱ͍͏ϧʔϧΛ༻͍Δͱɺମ Q(α, β) ͷ্Ͱ g(x) ͸࣍ͷΑ͏ʹҼ਺෼ղ͞ΕΔ͜ͱ
    ͕ɺ௚઀ͷܭࢉͰ֬ೝͰ͖·͢ɻ
    g(x) = (x − β)(x + α + β)
    ैͬͯɺf(x) ͸ɺQ(α, β) ͷ্Ͱɺ࣍ͷΑ͏ʹҼ਺෼ղ͞ΕΔ͜ͱʹͳΓ·͢ɻ
    f(x) = (x − α)(x − β)(x + α + β)

    ྫ 6-1 ͷܭࢉ͸ɺෳૉ਺ମ C ͷൣғͰߟ͑ͨࡍʹɺ1 ͷෳૉ 3 ৐ࠜͷ 1 ͭΛ ω ͱͯ͠ɺ
    α = 3

    2, β = ω 3

    2
    ͱஔ͍ͨ৔߹ͷܭࢉʹ૬౰͠·͢ɻͭ·Γɺ༗ཧ਺ମ Q ʹෳૉ਺ମͷݩ 3

    2, ω 3

    2 Λૉ๿ʹ෇͚Ճ͑ͯಘΒ
    ΕΔ෦෼ମΛ
    Q( 3

    2, ω 3

    2) ⊂ C
    ͱͯ͠ɺ
    Q(α, β) ∼
    = Q( 3

    2, ω 3

    2) (47)
    ͱ͍͏ಉ஋ؔ܎͕੒Γཱͭ͜ͱʹͳΓ·͢ɻ
    ͜ͷΑ͏ʹɺෳૉ਺ମͷଘࡏΛԾఆ͠ͳͯ͘΋ɺͦΕͱಉ݁͡Ռ͕ಘΒΕΔ఺͕ఆཧ 14 ͷϙΠϯτʹͳΔ
    Θ͚Ͱ͕͢ɺ(47) ͷΑ͏ͳؔ܎͸ɺৗʹ੒Γཱͭͱݴ͑ΔͷͰ͠ΐ͏͔ʁɹݴ͍׵͑Δͱɺෳૉ਺ମͷଘࡏΛ
    Ծఆͤͣʹఆཧ 14 ͷख๏Ͱ֦େମΛߏ੒ͨ࣌͠ʹɺෳૉ਺ମͷ෦෼ମͱಉܕʹͳΒͳ͍Α͏ͳಛผͳମ͕ಘ
    ΒΕΔՄೳੑ͸ͳ͍ͷͰ͠ΐ͏͔ʁ —— ݁࿦͔Βݴ͏ͱɺͦͷΑ͏ͳ͜ͱ͸͋Γ·ͤΜɻҰൠʹɺ͋Δମ F
    ʹ͍ͭͯɺF ্ͷ೚ҙͷଟ߲ࣜΛ 1 ࣍ࣜͷੵʹ෼ղՄೳʹ͢ΔΑ͏ͳ֦େମΛ F ͷ୅਺తดแͱݺͼɺ͜Ε
    33

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  34. ͸Ұҙʹఆ·Δ͜ͱ͕஌ΒΕ͍ͯ·͢ɻ༗ཧ਺ମ Q ʹରͯ͠͸ɺෳૉ਺ମ C ͕୅਺తดแʹͳΔ͜ͱ͕୅
    ਺ֶͷجຊఆཧͱͯ͠஌ΒΕ͍ͯ·͢ͷͰɺ্هͷख๏Ͱ֦େͨ͠ମ͸ɺෳૉ਺ମͷ෦෼ମͱಉܕʹͳΔ͜ͱ
    ͕อূ͞Ε·͢ɻຊߘͰ͸ɺ͜ͷࣄ࣮ͦͷ΋ͷ͸ূ໌͠·ͤΜ͕ɺগͳ͘ͱ΋ɺෳ਺ͷ෼ղମ͕ಘΒΕͨ৔߹
    ʹɺ͜ΕΒ͕ಉܕʹͳΔ͜ͱΛޙ΄Ͳʮ6.3 ࠷খ෼ղମͷҰҙੑʯͰূ໌͠·͢ɻ
    6.2 ࠷খ෼ղମͱΨϩΞ֦େ
    લઅͰಋೖͨ͠࠷খ෼ղମʹ͍ͭͯɺ࠷খ෼ղମͱͯ͠ಘΒΕΔ֦େମ͸ɺΨϩΞ֦େͱͳΓɺ͞Βʹɺ೚
    ҙͷΨϩΞ֦େ͸ɺ͋Δଟ߲ࣜʹର͢Δ࠷খ෼ղମͰ͋Δͱ͍͏ஶ͍͠ಛ௃͕͋Γ·͢ɻ͜͜Ͱ͸ɺ͜ͷࣄ࣮
    ΛॱΛ௥͍͖ͬͯࣔͯ͠·͢ɻ
    ·ͣɺલઅͰ͸ɺ֦େମ E ͷଘࡏΛԾఆͤͣʹɺଟ߲ࣜΛ෼ղՄೳʹ͢Δ֦େମ F(α) Λߏ੒͠·͕ͨ͠ɺ
    ͜͜ͰվΊͯɺ
    E ⊃ F(α) ⊃ F
    ͱͳΔ֦େମ E ͕ଘࡏ͢Δ৔߹Λߟ͑·͢ɻ͜͜ͰɺF(α) ͸ن໿ଟ߲ࣜ p(x) Λ༻͍֦ͯେͨ͠΋ͷͱ͠·
    ͢ɻͭ·ΓɺF(α) ্Ͱɺ
    p(x) = (x − α)g(x)
    ͱ͍͏Ҽ਺෼ղ͕ՄೳʹͳΓ·͢ɻҰํɺࠓͷ৔߹ɺα Λ E ͷݩͱΈͳͤ͹ɺ͜Ε͸ɺF ্Ͱ୅਺తͳݩͰ
    ͋Γɺͦͷ࠷খଟ߲ࣜ Irr(α, F) ͸ p(x) Ͱ༩͑ΒΕΔ͜ͱʹͳΓ·͢ɻͨͩ͠ɺE ͸ɺF(α) ΑΓ΋େ͖ͳ
    ମͳͷͰɺE ্Ͱߟ͑Ε͹ɺp(x) ͸ α Ҏ֎ʹ΋ࠜΛ࣋ͭՄೳੑ͕͋Γ·͢ɻ༰қʹ෼͔ΔΑ͏ʹɺ४ಉܕࣸ
    ૾ σ ∈ HomF
    (F(α), E) Λ༻͍Δͱɺσ(α) ͸࠶ͼ p(x) ͷࠜʹͳΓ·͢ɻ࣮ࡍɺ
    p(x) = xn + an−1
    xn−1 + · · · + a0
    (a0
    , · · · , an−1
    ∈ F)
    ͱ͢Δͱɺσ ͕ F ͷݩΛಈ͔͞ͳ͍४ಉܕࣸ૾Ͱ͋Δ͜ͱ͔Βɺ
    p(σ(α)) = σ(α)n + an−1
    σ(α)n−1 + · · · + a0
    = σ(αn + an−1
    αn−1 + · · · + a0
    )
    = σ(p(α)) = σ(0) = 0
    ͕੒Γཱͪ·͢ɻ͜ͷ࣌ɺHomF
    (F(α), E) ʹଐ͢Δ͢΂ͯͷࣸ૾Λ༻͍Δͱɺp(x) ͷ͢΂ͯͷ͕ࠜಘΒΕΔ
    ͱ͍͏ͷ͕ɺ࣍ͷิ୊ʹͳΓ·͢ɻ
    ิ୊ 6 ମͷ֦େ E/F ʹ͓͍ͯɺα ∈ E Λ F ্Ͱ୅਺తͳݩͱ͢Δɻ࠷খଟ߲ࣜ Irr(α, F) ͷ E ্ʹ͓͚
    Δ͢΂ͯͷࠜΛ
    A = {α1
    , · · · , αr
    } (α1
    = α)
    ͱ͢Δ࣌ɺ࣍͸શ୯ࣹͷࣸ૾Λ༩͑Δɻ
    F : HomF
    (F(α), E) −→ A
    σ −→ σ(α)
    ʢূ໌ʣ
    34

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  35. ࠷খଟ߲ࣜΛ p(x) ͱදه͢Δͱɺ೚ҙͷ σ ∈ HomF
    (F(α), E) ʹରͯ͠ɺσ(α) ∈ E ͸ p(σ(α)) = 0 Λຬ
    ͨ͢͜ͱ͔Βɺࣸ૾ F ͸ well-defined Ͱ͋Δɻ࣍ʹɺ࠷খଟ߲ࣜͷ࣍਺Λ n ͱͯ͠ɺF(α) ͷ೚ҙͷݩ͸ɺ
    x =
    n−1
    i=0
    ai
    αi (a0
    , · · · , an−1
    ∈ F) (48)
    ͱॻ͚ͯɺ͜ͷ x ʹରͯ͠ɺ
    σ(x) =
    n−1
    i=0
    ai
    σ(α)i
    ͕੒ཱ͢Δɻैͬͯɺσ1
    , σ2
    ∈ HomF
    (F(α), E) ʹ͍ͭͯɺ
    σ1
    (α) = σ2
    (α)
    Ͱ͋Ε͹ɺ
    ∀x ∈ F(α); σ1
    (x) = σ2
    (x)
    ͕੒ཱ͢Δɻ͜Ε͸ɺσ1
    = σ2
    Λҙຯ͓ͯ͠Γɺࣸ૾ F ͸୯ࣹͱͳΔɻҰํɺ೚ҙͷ αk
    ∈ A ʹରͯ͠ɺ(48)
    ͷ x Λ༻͍ͯɺࣸ૾ τ Λ
    τ(x) =
    n−1
    i=0
    ai
    αi
    k
    Ͱఆٛ͢Δͱɺ͜Ε͸ɺF ͷݩΛಈ͔͞ͳ͍४ಉܕࣸ૾Ͱ͋Γɺτ ∈ HomF
    (F(α), E) Ͱ͋Δ͜ͱ͕֬ೝͰ͖
    Δɻ͜ͷ࣌ɺτ(α) = αk
    ͱͳΔͷͰɺF ͸શࣹͰ͋Δɻ ˙
    ࣍ͷิ୊͸ɺن໿ଟ߲ࣜ͸ॏࠜΛ࣋ͨͳ͍ͱ͍͏ɺଟ߲ࣜͷجຊతͳੑ࣭Λࣔ͢΋ͷͰ͢ɻ
    ิ୊ 7 ମ F ্ͷط໿ଟ߲ࣜ f(x) ∈ F[x] ͸ɺͲͷΑ͏ͳ֦େମ E ʹ͓͍ͯ΋ॏࠜΛ࣋ͨͳ͍*10ɻ
    ʢূ໌ʣ
    f(x) ͷࠜͷ 1 ͭΛ α ∈ E ͱͯ͠ɺͦͷ࠷খଟ߲ࣜΛ p(x) ͱ͢Δɻf(x) Λ p(x) Ͱׂͬͨ༨ΓΛ r(x) ͱ
    ͢Δͱɺ
    f(x) = p(x)g(x) + r(x) (g(x), r(x) ∈ F[x])
    ͱॻ͚Δɻ͜ͷ࣌ɺf(α) = r(α) = 0 ͱͳΔ͕ɺr(x) ͸ p(x) ΑΓ΋࣍਺͕௿͍ͷͰɺp(x) ͕࠷খଟ߲ࣜͰ
    ͋Δͱ͍͏લఏ͔Βɺr(x) ͸߃౳తʹ 0 ʹͳΓɺ
    f(x) = p(x)g(x)
    ͕ಘΒΕΔɻ͞Βʹɺf(x) ͸ن໿ͳͷͰɺg(x) ͸ఆ਺ͱͳΓɺ݁ہɺ
    f(x) = ap(x) (a ∈ F)
    ͕ಘΒΕΔɻ
    ͜͜Ͱɺα ͕ f(x) ͷॏࠜͰɺ
    f(x) = (x − α)2h(x) (h(x) ∈ E[x])
    *10 ͜ͷิ୊͕੒ཱ͢Δʹ͸ɺମ F ͷඪ਺͕ 0 Ͱ͋Δͱ͍͏৚͕݅෇͖·͢ɻূ໌ͷதͰಋؔ਺ f′(x) Λ༻͍͍ͯ·͕͢ɺඪ਺͕ 0
    Ͱͳ͍ମͷ৔߹ɺಋؔ਺͕߃౳తʹ 0 ʹͳΔ৔߹͕͋ΔͷͰɺ͜ͷূ໌͸ద༻Ͱ͖·ͤΜɻ
    35

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  36. ͱ෼ղ͞ΕΔͱԾఆ͢Δͱɺf(x) ͷಋؔ਺ f′(x) ͸ɺ
    f′(x) = 2(x − α)h(x) + (x − α)2h′(x)
    ͱͳΓɺf′(α) = 0 Λຬ͕ͨ͢ɺf′(x) ͸ f(x) = ap(x) ΑΓ΋࣍਺͕௿͍ͷͰɺ͜Ε͸ɺp(x) ͕࠷খଟ߲ࣜ
    ͱ͍͏લఏʹໃ६͢Δɻैͬͯɺf(x) ͸ॏࠜΛ࣋ͨͳ͍ɻ ˙
    ࠷ޙʹɺ࠷খ෼ղମͷಛผͳੑ࣭ͱͯ͠ɺͦͷதؒମʹ͍ͭͯ੒Γཱͭิ୊Λࣔ͠·͢ɻ
    ิ୊ 8 ମ F ্ͷଟ߲ࣜ f(x) ∈ F[x] ͷ࠷খ෼ղମΛ E ͱ͢Δɻ֦େ E/F ͷ೚ҙͷதؒମ M ʹ͍ͭͯɺ
    ࣍ͷࣸ૾͸શࣹΛ༩͑Δɻ
    Aut(E/F) −→ HomF
    (M, E)
    σ −→ σ|M
    ͭ·ΓɺHomF
    (M, E) ʹଐ͢Δ೚ҙͷࣸ૾͸ɺAut(E/F) ʹଐ͢Δࣸ૾ͷఆٛҬΛ M ʹ੍ݶͨ͠΋ͷͱ͠
    ͯಘΒΕΔɻ
    ʢূ໌ʣ
    ೚ҙͷ τ ∈ HomF
    (M, E) ʹରͯ͠ɺτ ͕४ಉܕࣸ૾Ͱ͋Δ͜ͱ͔Βɺͦͷ૾
    M′ = τ(M)
    ͸ମʹͳΔɻ͞ΒʹɺମΛఆٛҬͱ͢Δ४ಉܕࣸ૾ͳͷͰ୯ࣹͰ͋Δ఺ʹ஫ҙ͢Δͱɺτ ͸ɺM ͔Β M′ ͷ
    ಉܕࣸ૾Λ༩͑Δ͜ͱ͕෼͔Δɻ࣍ʹɺf(x) ͷ࣍਺Λ n ͱͯ͠ɺE ্Ͱɺ
    f(x) = a(x − α1
    )(x − α2
    ) · · · (x − αn
    ) (a ∈ F) (49)
    ͱҼ਺෼ղ͞ΕΔͱ͢Δͱɺ࠷খ෼ղମͷఆٛΑΓɺ
    E = F(α1
    , · · · , αn
    )
    ͕੒ཱ͢Δɻࠓɺ{α1
    , · · · , αn
    } ͕͢΂ͯ M ʹؚ·Ε͍ͯΕ͹ɺͦΕ͸ E = M Λҙຯ͢ΔͷͰɺ
    HomF
    (M, E) = Aut(E/F)
    ͱͳΓɺิ୊ͷओு͸ࣗ໌ͱͳΔɻͦ͜ͰɺࠓɺM ʹؚ·Ε͍ͯͳ͍΋ͷΛ {α1
    , · · · , αr
    } ͱ͢Δɻ͜ͷ࣌ɺ
    M1
    = M(α1
    ), M2
    = M(α1
    , α2
    ), · · · , Mr
    = M(α1
    , · · · , αr
    )
    M′
    1
    = M′(α′
    1
    ), M′
    2
    = M′(α′
    1
    , α′
    2
    ), · · · , M′
    r
    = M′(α′
    1
    , · · · , α′
    r
    )
    ͱͯ͠ɺҰ࿈ͷࣸ૾Λ࣍ͷΑ͏ʹؼೲతʹఆٛ͢Δɻ
    τ0
    ∈ HomF
    (M, M′) : τ0
    (x) = τ(x)
    τ1
    ∈ HomF
    (M1
    , M′
    1
    ) : τ1
    (α1
    ) = α′
    1
    , τ1
    (x) = τ0
    (x) (x ∈ M)
    τ2
    ∈ HomF
    (M2
    , M′
    2
    ) : τ2
    (α2
    ) = α′
    2
    , τ2
    (x) = τ1
    (x) (x ∈ M1
    )
    .
    .
    .
    τk
    ∈ HomF
    (Mk
    , M′
    k
    ) : τk
    (αk
    ) = α′
    k
    , τk
    (x) = τk−1
    (x) (x ∈ Mk−1
    )
    .
    .
    .
    τr
    ∈ HomF
    (Mr
    , M′
    r
    ) : τr
    (αr
    ) = α′
    r
    , τr
    (x) = τr−1
    (x) (x ∈ Mr−1
    )
    36

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  37. ͜͜Ͱɺ{α′
    1
    , · · · , α′
    r
    } ͸ɺ{α1
    , · · · αr
    } ͷॱ൪Λฒ΂ସ͑ͨ΋ͷͰɺ͜ͷબ୒Λ͏·͘ߦ͏ͱɺ্هͷࣸ૾͕
    ͢΂ͯશ୯ࣹͷ४ಉܕࣸ૾ɺͭ·Γɺମͱͯ͠ͷಉܕࣸ૾ʹͳΔ͜ͱΛ k (0 ≤ k ≤ r) ʹ͍ͭͯͷ਺ֶతؼೲ
    ๏Ͱূ໌͢Δɻk = 0 ͷ࣌͸ࣗ໌ͳͷͰɺk − 1 ·Ͱ੒ཱ͍ͯ͠Δͱͯ͠ɺk (k ≥ 1) ͷ৔߹Λߟ͑Δɻ
    ·ͣɺଟ߲ࣜ g(x) ∈ Mk−1
    [x] ʹରͯ͠ɺ͢΂ͯͷ܎਺ʹ τk−1
    Λ࡞༻ͨ͠΋ͷΛ gτk−1 (x) ∈ M′
    k−1
    [x] ͱ
    දه͢Δͱɺτk−1
    ͕ಉܕࣸ૾ͱ͍͏Ծఆ͔Βɺ࣍͸ɺଟ߲ࣜ؀ͷؒͷʢ؀ͱͯ͠ͷʣಉܕࣸ૾ͱͳΔɻ
    τk−1
    : Mk−1
    [x] −→ M′
    k−1
    [x]
    g(x) −→ gτk−1 (x)
    ࣍ʹɺαk
    ͷ Mk−1
    ্ͷ࠷খଟ߲ࣜΛ p(x) ͱͯ͠ʢαk
    /
    ∈ Mk−1
    Ͱ͋Δ͜ͱ͔Βɺp(x) ͸ 2 ࣍Ҏ্Ͱ͋Δ
    ͜ͱʹ஫ҙ͢Δʣ
    ɺf(x) Λ Mk−1
    ্ͷଟ߲ࣜͱׂͯͬͨ࣌͠ͷ঎Λ q(x) ͱͯ͠ɺ
    f(x) = p(x)q(x) (p(x), q(x) ∈ Mk−1
    [x])
    ͱදΘ͢ɻ͜͜Ͱɺp(αk
    ) = f(αk
    ) = 0 ΑΓɺ৒༨߲͸߃౳తʹ 0 ʹͳΔ͜ͱΛ༻͍ͨɻ͜ͷ྆ลʹ τk−1
    Λ
    ࡞༻ͤ͞ΔͱɺM′
    k−1
    ্ͷଟ߲ࣜͷؔ܎ͱͯ͠ɺ
    f(x) = pτk−1 (x)qτk−1 (x) (pτk−1 (x), qτk−1 (x) ∈ M′
    k−1
    [x])
    ͕ಘΒΕΔɻτk−1
    ͸ F ͷݩ͸ಈ͔͞ͳ͍ͷͰɺfτk−1 (x) = f(x) ͱͳΔࣄΛ༻͍ͨɻ͜͜Ͱɺpτk−1 (x) ͸ 2
    ࣍Ҏ্ͷଟ߲ࣜͳͷͰɺ(49) ͷগͳ͘ͱ΋ 2 ͭͷҼ਺͕ pτk−1 (x) ʹؚ·Ε͓ͯΓɺ
    pτk−1 (α′
    k
    ) = 0
    ͱͳΔ α′
    k
    ∈ {α1
    , · · · αn
    } ͕બ୒Ͱ͖Δɻͦͯ͠ɺ͜ͷ࣌ɺα′
    k
    /
    ∈ M′
    k−1
    Ͱʢ͜ͷ৚݅ΑΓɺα′
    k
    ∈ {α1
    , · · · αr
    }
    Ͱ͋Γɺ͔ͭɺಉ͡ αi
    Λॏෳͯ͠બͿ͜ͱ͸ͳ͍ʣ
    ɺpτk−1 (x) ͸ɺα′
    k
    ʹର͢Δ M′
    k−1
    ্ͷ࠷খଟ߲ࣜͰ͋Δ
    ͜ͱ͕ࣔͤΔɻ·ͣɺpτk−1 (x) ∈ M′
    k−1
    [x] ͸ɺ
    ʢ࠷ߴ࣍਺ͷ܎਺Λ 1 ͱ͢Δʣن໿ଟ߲ࣜ pk−1
    (x) ∈ Mk−1
    [x]
    Λ Mk−1
    [x] ͔Β M′
    k−1
    [x] ΁ͷଟ߲ࣜ؀ͷಉܕࣸ૾ τk−1
    ͰҠͨ͠΋ͷͰ͋Δ͔ΒɺM′
    k−1
    ্ͷʢ࠷ߴ࣍਺ͷ
    ܎਺Λ 1 ͱ͢Δʣن໿ଟ߲ࣜͰ͋Δɻͦͯ͠ɺα′
    k
    ͸ 2 ࣍Ҏ্ͷن໿ଟ߲ࣜ pτk−1 (x) ͷࠜͰ͋Δ͔Β M′
    k−1
    ͷݩͰ͸ͳ͍ɻ
    ʢ͞΋ͳ͘͹ɺpτk−1 (x) ͕ x − α′
    k
    ΛҼ਺ʹ࣋ͪɺن໿Ͱͳ͘ͳΔʣ
    ɻ·ͨɺن໿Ͱ͋Δ͜ͱ͔
    Β࠷খ࣍਺Ͱ͋Δ͜ͱ΋ݴ͑ΔͷͰɺpτk−1 (x) ͸࠷খଟ߲ࣜͱͳΔɻ
    ैͬͯɺ࠷খଟ߲ࣜ pτk−1 (x) Λ༻͍ͯɺ৒༨ମ M′
    k−1
    [x]/pτk−1 (x) Λߏ੒͢Δ͜ͱ͕Ͱ͖ͯɺ͜Ε͸ɺ
    M′
    k
    = M′
    k−1
    (α′
    k
    ) ͱಉܕʹͳΔɻ͜Εͱಉ༷ʹɺ৒༨ମ Mk−1
    [x]/p(x) ͸ɺମ Mk
    = Mk−1
    (αk
    ) ͱಉܕʹͳ
    ΔͷͰɺMk
    ͔Β M′
    k
    ΁ͷࣸ૾Λ৒༨ମͷؒͷࣸ૾ͱͯ͠ɺ࣍ͷΑ͏ʹఆٛ͢Δ͜ͱ͕Ͱ͖Δɻ
    τk
    : Mk

    = Mk−1
    [x]/p(x) −→ M′
    k

    = M′
    k−1
    [x]/pτk−1 (x)
    g(x)/p(x) −→ gτk−1 (x)/pτk−1 (x)
    τk−1
    ͕ଟ߲ࣜ؀ Mk−1
    [x] ͱ M′
    k−1
    [x] ͷؒͷಉܕࣸ૾Ͱ͋Δ͜ͱ͔Βɺ͜Ε͸৒༨ମͱͯ͠ͷಉܕࣸ૾Λ༩
    ͍͑ͯΔ͜ͱ͕෼͔Δɻ͞Βʹɺ͜ͷࣸ૾͸ɺMk−1
    (αk
    ) ͱ M′
    k−1
    (α′
    k
    ) ͷݩͰදݱ͢Δͱɺ๯಄Ͱఆٛͨ͠
    τk
    ʹҰக͢ΔɻΑͬͯɺ਺ֶతؼೲ๏ʹΑΓɺಉܕࣸ૾ τr
    ∈ HomF
    (Mr
    , M′
    r
    ) ͕ఆٛ͞Εͨɻ
    ࠷ޙʹɺMr
    = E, M′
    r
    = E ͱͳΔ͜ͱʹ஫ҙ͢Δͱɺ
    HomF
    (Mr
    , M′
    r
    ) = Aut(E/F)
    ͱͳΔ͜ͱ͔Βɺτr
    ∈ Aut(E/F) Ͱ͋Γɺ͞Βʹલड़ͷఆٛΑΓɺτr
    |M
    = τ ͱͳΔ͜ͱ͕෼͔ΔͷͰɺ͜Ε
    Ͱิ୊ͷओு͕ূ໌͕ࣔ͞Εͨɻ ˙
    37

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  38. ͦΕͰ͸ɺҎ্ͷ४උͷԼʹɺΨϩΞ֦େ E/F ͸ɺE ͕ F ্ͷଟ߲ࣜͷ෼ղମͰ͋Δ͜ͱͱಉ஋Ͱ͋Δ
    ͱ͍͏࣍ͷఆཧΛࣔ͠·͢ɻ
    ఆཧ 15 ༗ݶ࣍ݩͷ֦େ E/F ͕ΨϩΞ֦େͰ͋Δ͜ͱ͸ɺ࣍ͷͦΕͧΕͷ৚݅ͱಉ஋Ͱ͋Δɻ
    (a) E ͸͋Δଟ߲ࣜ f(x) ∈ F[X] ͷ E ্ͷࠜ {α1
    , · · · , αn
    } Λ༻͍ͯɺ
    E = F(α1
    , · · · , αn
    )
    ͱද͞ΕΔɻͭ·ΓɺE ͸ f(x) ͷ F ্ͷ࠷খ෼ղମͰ͋Δɻ
    (b) ೚ҙͷن໿ଟ߲ࣜ p(x) ∈ F[x] ͕ E ʹࠜΛ࣋ͭ৔߹ɺp(x) ͸ɺE ্Ͱ͢΂ͯͷࠜʹର͢ΔҰ࣍ࣜͷੵʹ
    Ҽ਺෼ղ͞ΕΔɻͭ·ΓɺE ͸ p(x) ͷ F ্ͷ෼ղମͰ͋Δɻ
    ʢূ໌ʣ
    ɾE/F ͕ΨϩΞ֦େ ⇒ (b)
    p(x) ͷ E ʹ͓͚Δ͢΂ͯͷࠜΛ A = {α1
    , · · · , αr
    } ͱͯ͠ɺ࣍ͷଟ߲ࣜΛߟ͑Δɻ
    q(x) =
    r
    i=1
    (x − αi
    ) ∈ E[x] (50)
    (50) ͷӈลΛల։ͨ࣌͠ʹಘΒΕΔ֤߲ͷ܎਺͸ɺA ͷཁૉͷରশͳ૊Έ߹ΘͤͰɺ͜ΕΒͷཁૉͷஔ׵ʹ
    ରͯ͠ෆมͰ͋Δɻ
    ҰํɺG = Aut(E/F) ͱͯ͠ɺ೚ҙͷ σ ∈ G ʹରͯ͠ɺ
    p(σ(αi
    )) = σ(p(αi
    )) = 0 (i = 1, · · · , r)
    ͱͳΔࣄ͔ΒɺͦΕͧΕͷ σ(αi
    ) ͸ A ͷཁૉͷͲΕ͔ʹҰகͯ͠ɺ͔ͭɺσ ͕શ୯ࣹͰ͋Δ͜ͱΛߟ͑Δͱɺ
    σ ͸ɺू߹ A ͷஔ׵ΛҾ͖ى͜͢͜ͱʹͳΔɻैͬͯɺલड़ͷల։܎਺͸ɺG ʹଐ͢Δࣸ૾ʹΑͬͯಈ͔ͳ
    ͍ EG ͷݩͰ͋Γɺఆཧ 6 ͔Β F = EG Ͱ͋ΔͷͰɺq(x) ͷӈลΛల։ͨ͠΋ͷ͸ɺ
    q(x) ∈ F[x]
    ͱͳΔ͜ͱ͕෼͔Δɻͦ͜ͰɺF[x] ͷൣғͰ p(x) Λ q(x) Ͱׂͬͯɺ
    p(x) = q(x)g(x) + r(x)
    ͱදΘ͢ͱɺp(αi
    ) = q(αi
    ) = 0 ΑΓɺr(x) ͸ A ͷ͢΂ͯͷཁૉΛࠜʹ͕࣋ͭɺr(x) ͷ࣍਺Λߟ͑Δͱ r(x)
    ͸߃౳తʹ 0 ʹͳΔɻ͞Βʹɺp(x) ͕ن໿Ͱ͋Δ͜ͱ͔Βɺg(x) ͸ఆ਺ʹͳΓɺ
    p(x) = cq(x) (c ∈ F)
    ͕੒ཱ͢Δɻq(x) ͸ (50) ͷΑ͏ʹ E ʹ͓͍ͯҼ਺෼ղ͞ΕΔͷͰɺ͜ΕͰओு͕ࣔ͞Εͨɻ
    ɾ(b) ⇒ (a)
    E/F ͸༗ݶ࣍ݩͷ֦େͳͷͰɺE ͷ༗ݶݸͷݩ {α1
    , · · · , αr
    } Λ༻͍ͯɺ
    E = F(α1
    , · · · , αr
    )
    38

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  39. ͱॻ͚Δ͜ͱ͕ࣔͤΔɻͨͱ͑͹ɺα1
    ∈ E (α1
    /
    ∈ F) Λ༻͍ͯ E1
    = F(α1
    ) ͱͨ࣌͠ʹɺE1
    ͸ E ʹҰக͠
    ͳ͔ͬͨͱ͢Δɻ͜ͷ࣌ɺα2
    ∈ E (α2
    /
    ∈ E1
    ) ΛऔΔͱɺα1
    ͱ α2
    ͸ɺF ্ͷϕΫτϧۭؒ E ʹ͓͍ͯҰ࣍
    ಠཱͰ͋Δɻͦ͜ͰɺE2
    = F(α1
    , α2
    ) ͱͯ͠ɺ͞ΒʹɺE2
    ͕ E ʹҰக͠ͳ͔ͬͨͱ͢Δͱɺಉ༷ʹͯ͠Ұ
    ࣍ಠཱͳݩ α3
    ͕औΕΔɻ͜ΕΒΛ܁Γฦͨ͠ࡍʹɺ༗ݶճͰ En
    = E ͱͳΒͳ͔ͬͨ৔߹ɺແݶݸͷҰ࣍ಠ
    ཱͳݩ͕ଘࡏ͢Δ͜ͱʹͳΓɺE/F ͕༗ݶ࣍ݩͷ֦େͰ͋Δ͜ͱʹໃ६͢Δɻ
    ࣍ʹɺܥ 1 ΑΓɺ೚ҙͷ αi
    ͸ɺF ্ͷ࠷খଟ߲ࣜ pi
    (x) Λ࣋ͪɺ(b) ͷԾఆΑΓɺE ্ͰҰ࣍ࣜͷੵʹҼ
    ਺෼ղ͞ΕΔɻͦ͜Ͱɺ
    f(x) =
    r
    i=1
    pi
    (x)
    ͱ͍͏ F ্ͷଟ߲ࣜΛߟ͑Δͱɺ͜Ε͸ (a) ͷ৚݅Λຬͨ͢ଟ߲ࣜͱͳΔɻ
    ɾ(a) ⇒ E/F ͕ΨϩΞ֦େ
    E ͸͋Δଟ߲ࣜ f(x) ͷ࠷খ෼ղମͰ͋Δ͜ͱ͔Βɺิ୊ 8 ͕ར༻Ͱ͖Δɻ·ͨɺఆཧ 6 ΑΓɺ
    EAut(E/F ) = F (51)
    ͕ࣔͤΕ͹Α͍ɻ
    Ұൠʹ͸ɺEAut(E/F ) ⊇ F Ͱ͋Δ͜ͱʹ஫ҙͯ͠ɺ೚ҙͷ α ∈ EAut(E/F ) ʹରͯ͠தؒମ F(α) Λߟ͑
    Δͱɺ
    EAut(E/F ) ⊇ F(α) ⊇ F
    ͕੒Γཱͭɻ͜ͷ࣌ɺα ∈ EAut(E/F ) ͸ Aut(E/F) ʹΑͬͯಈ͔ͳ͍ࣄΛߟ͑Δͱɺ೚ҙͷ σ ∈ Aut(E/F)
    ͸ɺఆٛҬΛ F(α) ʹ੍ݶ͢Δͱ߃౳ࣸ૾ʹͳΔ͜ͱ͕෼͔Δɻैͬͯɺิ୊ 8 ΑΓɺHomF
    (F(α), E) ͸߃
    ౳ࣸ૾ͷΈΛؚΈɺ
    |HomF
    (F(α), E)| = 1 (52)
    ͱͳΔɻ
    Ұํɺܥ 1 ΑΓ α ͸ F ্ͷ࠷খଟ߲ࣜ p(x) ∈ F[x] ͓࣋ͬͯΓɺͦͷ͢΂ͯͷࠜΛ A = {α1
    , · · · , αr
    } ͱ
    ͢Δͱɺิ୊ 6 ΑΓɺHomF
    (F(α), E) ͱ A ͸ू߹ͱͯ͠ 1 ର 1 ʹͳΔɻैͬͯɺ(52) ΑΓ p(x) ͷࠜ͸ α
    ͷΈͰɺิ୊ 7 ΑΓ p(X) ͕ॏࠜΛ࣋ͭ͜ͱ͸ͳ͍͜ͱ͔Βɺ
    p(x) = x − α
    ͕ಘΒΕΔɻैͬͯɺα ∈ F Ͱͳ͚Ε͹ͳΒͣɺ(51) ͕ࣔ͞Εͨɻ ˙
    6.3 ࠷খ෼ղମͷҰҙੑ
    ͜͜Ͱ͸ɺମ F ্ͷଟ߲ࣜ f(x) ʹରͯ͠ɺෳ਺ͷ࠷খ෼ղମ͕ߏ੒͞Εͨ৔߹ɺͦΕΒ͸ಉܕʹͳΔ͜ͱ
    Λࣔ͠·͢ɻ͜ΕʹΑΓɺఆཧ 14 ͷखଓ͖ʹैͬͯ࠷খ෼ղମ F(α1
    , · · · , αn
    ) Λߏ੒͢ΔࡍɺࠜΛ෇͚Ճ͑
    Δॱ൪ʹΑΒͣʹɺಉ͡࠷খ෼ղମ͕ಘΒΕΔ͜ͱ͕อূ͞Ε·͢ɻ࣍ͷఆཧͷূ໌Ͱ͸ɺิ୊ 8 ͷূ໌ͱ΄
    ΅ಉ͡ํ๏Λ༻͍ͯɺ۩ମతͳಉܕࣸ૾Λؼೲతʹߏ੒͠·͢ɻ
    39

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  40. ఆཧ 16 2 छྨͷ֦େ E/Fɺ͓ΑͼɺE′/F ͕͋ΓɺͲͪΒ΋ F ্ͷ n ࣍ଟ߲ࣜ f(x) ͷ࠷খ෼ղମʹͳͬ
    ͓ͯΓɺ
    E = F(α1
    , · · · , αn
    ), f(x) = a
    n
    i=1
    (x − αi
    )
    E′ = F(β1
    , · · · , βn
    ), f(x) = b
    n
    i=1
    (x − βi
    )
    ͕੒Γཱͭ΋ͷͱ͢Δɻ͜ͷ࣌ɺE ͔Β E′ ΁ͷಉܕࣸ૾Ͱɺ{α1
    , · · · , αn
    } Λ {β1
    , · · · , βn
    } ʹஔ׵͢Δ΋ͷ
    ͕ߏ੒Ͱ͖Δɻ
    ʢূ໌ʣ
    F1
    = F(α1
    ), F2
    = F(α1
    , α2
    ), · · · , Fn
    = F(α1
    , · · · , αn
    )
    F′
    1
    = F′(β′
    1
    ), F′
    2
    = F′(β′
    1
    , β′
    2
    ), · · · , F′
    n
    = F′(β′
    1
    , · · · , β′
    n
    )
    ͱͯ͠ɺҰ࿈ͷࣸ૾Λ࣍ͷΑ͏ʹؼೲతʹఆٛ͢Δɻ
    τ0
    ∈ HomF
    (F, F) : τ0
    (x) = x
    τ1
    ∈ HomF
    (F1
    , F′
    1
    ) : τ1
    (α1
    ) = β′
    1
    , τ1
    (x) = τ0
    (x) (x ∈ F)
    τ2
    ∈ HomF
    (F2
    , F′
    2
    ) : τ2
    (α2
    ) = β′
    2
    , τ2
    (x) = τ1
    (x) (x ∈ F1
    )
    .
    .
    .
    τk
    ∈ HomF
    (Fk
    , F′
    k
    ) : τk
    (αk
    ) = β′
    k
    , τk
    (x) = τk−1
    (x) (x ∈ Fk−1
    )
    .
    .
    .
    τn
    ∈ HomF
    (Fn
    , F′
    n
    ) : τn
    (αr
    ) = β′
    n
    , τr
    (x) = τn−1
    (x) (x ∈ Fn−1
    )
    ͜͜Ͱɺ{β′
    1
    , · · · , β′
    n
    } ͸ɺ{β1
    , · · · , βn
    } ͷॱ൪Λฒ΂ସ͑ͨ΋ͷͰɺ͜ͷબ୒Λ͏·͘ߦ͏ͱɺ্هͷࣸ૾
    ͕͢΂ͯମͱͯ͠ͷಉܕࣸ૾ʹͳΔ͜ͱΛ k (0 ≤ k ≤ n) ʹ͍ͭͯͷ਺ֶతؼೲ๏Ͱূ໌͢Δɻk = 0 ͷ࣌͸
    ࣗ໌ͳͷͰɺk − 1 ·Ͱ੒ཱ͍ͯ͠Δͱͯ͠ɺk (k ≥ 1) ͷ৔߹Λߟ͑Δɻ
    ·ͣɺଟ߲ࣜ g(x) ∈ Fk−1
    [x] ʹରͯ͠ɺ͢΂ͯͷ܎਺ʹ τk−1
    Λ࡞༻ͨ͠΋ͷΛ gτk−1 (x) ∈ F′
    k−1
    [x] ͱද
    ه͢Δͱɺτk−1
    ͕ಉܕࣸ૾ͱ͍͏Ծఆ͔Βɺ࣍͸ɺଟ߲ࣜ؀ͷؒͷʢ؀ͱͯ͠ͷʣಉܕࣸ૾ͱͳΔɻ
    τk−1
    : Fk−1
    [x] −→ F′
    k−1
    [x]
    g(x) −→ gτk−1 (x)
    ࣍ʹɺαk
    ͷ Fk−1
    ্ͷ࠷খଟ߲ࣜ p(x) ͱͯ͠ʢαk
    /
    ∈ Fk−1
    Ͱ͋Δ͜ͱ͔Βɺp(x) ͸ 2 ࣍Ҏ্Ͱ͋Δ͜ͱ
    ʹ஫ҙ͢Δʣ
    ɺf(x) Λ Fk−1
    ্ͷଟ߲ࣜͱׂͯͬͨ࣌͠ͷ঎Λ q(x) ͱͯ͠ɺ
    f(x) = p(x)q(x) (p(x), q(x) ∈ Fk−1
    [x])
    ͱදΘ͢ɻ͜͜Ͱɺp(αk
    ) = f(αk
    ) = 0 ΑΓɺ৒༨߲͸߃౳తʹ 0 ʹͳΔ͜ͱΛ༻͍ͨɻ͜ͷ྆ลʹ τk−1
    Λ
    ࡞༻ͤ͞ΔͱɺF′
    k−1
    ্ͷଟ߲ࣜͷؔ܎ͱͯ͠ɺ
    f(x) = pτk−1 (x)qτk−1 (x) (pτk−1 (x), qτk−1 (x) ∈ F′
    k−1
    [x])
    ͕ಘΒΕΔɻτk−1
    ͸ F ͷݩ͸ಈ͔͞ͳ͍ͷͰɺfτk−1 (x) = f(x) ͱͳΔࣄΛ༻͍ͨɻ͜͜Ͱɺpτk−1 (x) ͸ 2
    ࣍Ҏ্ͷଟ߲ࣜͳͷͰɺE′ ্Ͱ f(x) ΛҼ਺෼ղͨ͠ࡍͷগͳ͘ͱ΋ 2 ͭͷҼ਺͕ pτk−1 (x) ʹؚ·Ε͓ͯΓɺ
    pτk−1 (β′
    k
    ) = 0
    40

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  41. ͱͳΔ β′
    k
    ∈ {β1
    , · · · βn
    } ͕બ୒Ͱ͖Δɻͦͯ͠ɺ
    ͜ͷ࣌ɺ
    β′
    k
    /
    ∈ F′
    k−1
    Ͱ
    ʢ͜ͷ৚݅ΑΓɺ
    ಉ͡ βi
    Λॏෳͯ͠બͿ
    ͜ͱ͸ͳ͍ʣ
    ɺ
    pτk−1 (x) ͸ɺ
    β′
    k
    ʹର͢Δ F′
    k−1
    ্ͷ࠷খଟ߲ࣜͰ͋Δ͜ͱ͕ࣔͤΔɻ·ͣɺ
    pτk−1 (x) ∈ F′
    k−1
    [x]
    ͸ɺ
    ʢ࠷ߴ࣍਺ͷ܎਺Λ 1 ͱ͢Δʣن໿ଟ߲ࣜ pk−1
    (x) ∈ Fk−1
    [x] Λ Fk−1
    [x] ͔Β F′
    k−1
    [x] ΁ͷଟ߲ࣜ؀ͷಉ
    ܕࣸ૾ τk−1
    ͰҠͨ͠΋ͷͰ͋Δ͔ΒɺF′
    k−1
    ্ͷʢ࠷ߴ࣍਺ͷ܎਺Λ 1 ͱ͢Δʣن໿ଟ߲ࣜͰ͋Δɻͦͯ͠ɺ
    β′
    k
    ͸ 2 ࣍Ҏ্ͷن໿ଟ߲ࣜ pτk−1 (x) ͷࠜͰ͋Δ͔Β F′
    k−1
    ͷݩͰ͸ͳ͍ɻ
    ʢ͞΋ͳ͘͹ɺpτk−1 (x) ͕ x − β′
    k
    ΛҼ਺ʹ࣋ͪɺن໿Ͱͳ͘ͳΔʣ
    ɻ·ͨɺن໿Ͱ͋Δ͜ͱ͔Β࠷খ࣍਺Ͱ͋Δ͜ͱ΋ݴ͑ΔͷͰɺpτk−1 (x) ͸
    ࠷খଟ߲ࣜͱͳΔɻ
    ैͬͯɺ࠷খଟ߲ࣜ pτk−1 (x) Λ༻͍ͯɺ৒༨ମ F′
    k−1
    [x]/pτk−1 (x) Λߏ੒͢Δ͜ͱ͕Ͱ͖ͯɺ͜Ε͸ɺ
    F′
    k
    = F′
    k−1
    (β′
    k
    ) ͱಉܕʹͳΔɻ͜Εͱಉ༷ʹɺ৒༨ମ Fk−1
    [x]/p(x) ͸ɺମ Fk
    = Fk−1
    (αk
    ) ͱಉܕʹͳΔͷ
    ͰɺFk
    ͔Β F′
    k
    ΁ͷࣸ૾Λ৒༨ମͷؒͷࣸ૾ͱͯ͠ɺ࣍ͷΑ͏ʹఆٛ͢Δ͜ͱ͕Ͱ͖Δɻ
    τk
    : Fk

    = Fk−1
    [x]/p(x) −→ F′
    k

    = F′
    k−1
    [x]/pτk−1 (x)
    g(x)/p(x) −→ gτk−1 (x)/pτk−1 (x)
    τk−1
    ͕ଟ߲ࣜ؀ Fk−1
    [x] ͱ F′
    k−1
    [x] ͷؒͷಉܕࣸ૾Ͱ͋Δ͜ͱ͔Βɺ͜Ε͸৒༨ମͱͯ͠ͷಉܕࣸ૾Λ༩͑
    ͍ͯΔ͜ͱ͕෼͔Δɻ͞Βʹɺ͜ͷࣸ૾͸ɺFk−1
    (αk
    ) ͱ F′
    k−1
    (β′
    k
    ) ͷݩͰදݱ͢Δͱɺ๯಄Ͱఆٛͨ͠ τk
    ʹ
    Ұக͢ΔɻΑͬͯɺ਺ֶతؼೲ๏ʹΑΓɺಉܕࣸ૾ τn
    ∈ HomF
    (Fn
    , F′
    n
    ) ͕ఆٛ͞Εͨɻ࠷ޙʹɺE = Fn
    ͓
    Αͼ E′ = F′
    n
    ͱͳΔ͜ͱ͔Βɺ͜Ε͕ٻΊΔಉܕࣸ૾Λ༩͑Δ͜ͱʹͳΔɻ ˙
    7 ୅਺ํఔࣜͷՄղ৚݅
    7.1 ΂͖֦ࠜେͱՄղ܈
    ͜͜Ͱ͸ɺಛʹɺଟ߲ࣜ xn − a ͷ࠷খ෼ղମ͕ຬͨ͢ੑ࣭Λௐ΂·͢ɻҰൠʹ xn − a ͷࠜΛ΂͖ࠜͱݺ
    ͼ·͕͢ɺ͜͜Ͱࣔ͢ੑ࣭͸ɺ͜ͷޙɺ୅਺ํఔࣜͷղ͕΂͖ࠜΛ༻͍ͯදݱͰ͖Δ͔Ͳ͏͔Λ൑ఆ͢ΔͨΊ
    ͷॏཁͳख͕͔ΓͱͳΓ·͢ɻ·ͣ͸ɺa = 1 ͷ৔߹ʹ͍ͭͯ੒Γཱͭิ୊Λࣔ͠·͢ɻ
    ิ୊ 9 ଟ߲ࣜ f(x) = xn − 1 ∈ F[x] ͸ॏࠜΛ࣋ͨͳ͍ͱ͢Δ*11ɻf(x) ͷ࠷খ෼ղମΛ E ͱ͢Δ࣌ɺ
    E = F(ω) ͱͳΔ ω ∈ E ͕ଘࡏͯ͠ɺAut(F(ω)/F) ͸Ξʔϕϧ܈ G0
    ͷ෦෼܈ʹಉܕͱͳΔɻ͜͜ͰɺG0
    ͸ɺZ Λ੔਺؀ͱͯ͠ɺ1 ≤ t < n ͷൣғͰ n ͱޓ͍ʹૉͳ t ∈ Z ΛूΊͨू߹ʹରͯ͠ɺZ/nZ ্Ͱͷੵ
    ΛೖΕͯ܈ʹͨ͠΋ͷͰ͋Δɻ·ͨɺω ͸ɺ1 ͷݪ࢝ n ৐ࠜʹͳ͓ͬͯΓɺ
    ωn = 1
    ωi = ωj (i = j)
    Λຬͨ͢ɻ
    ʢূ໌ʣ
    f(x) ͷ n ݸͷ૬ҧͳΔࠜͷू߹Λ A = {ζ ∈ E | ζn = 1} ⊆ E ͱ͢Δͱɺ͜Ε͸ɺE ͷੵʹؔͯ͠܈ͱͳ
    Δɻମʹؚ·ΕΔ༗ݶ෦෼܈͸ɺ८ճ܈ͱͳΔ͜ͱ͕஌ΒΕ͓ͯΓɺA = {1, ω, · · · , ωn−1} ͱͳΔ ω ∈ E ͕
    ଘࡏ͢Δɻ͜ͷ࣌ɺA ͷݩ͸͢΂ͯ ω ͔Βੜ੒͞ΕΔͷͰɺF(ω) ͸ f(x) ͷ࠷খ෼ղମͰ͋ΓɺE = F(ω)
    *11 ମ F ͕༗ཧ਺ମ Q ͷ৔߹ɺ୅਺ֶͷجຊఆཧʹΑͬͯɺ͜ͷ৚͕݅ຬͨ͞Ε·͢ɻΑΓҰൠʹ͸ɺମ F ͷඪ਺͕ 0 Ͱ͋Δ৔
    ߹ʹɺ͜ͷ৚͕݅ຬͨ͞Ε·͢ɻ͜ΕҎ߱ɺମ F ͷඪ਺͕ 0 Ͱ͋Δͱ͍͏ຊߘͷલఏ͸ಛʹ໌هͤͣʹɺิ୊ 9 Λར༻͍͖ͯ͠
    ·͢ɻ
    41

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  42. ͱͳΔɻ͋Δ͍͸ɺ೚ҙͷ t ∈ G0
    Λ 1 ͭબͿͱɺt ͱ n ͕ޓ͍ʹૉͰ͋Δ͜ͱ͔Βɺ೚ҙͷࣗવ਺ a ͱ b ʹ
    ͍ͭͯɺ
    at = bt (mod n) ⇔ a = b (mod n)
    ͕੒ཱ͢ΔͷͰɺ{1, ωt, ω2t, · · · , ω(n−1)t} ͸͢΂ͯҟͳΔݩͱͳΓɺ͜ΕΒ΋ A ʹҰக͢Δɻ
    A = {1, ωt, ω2t, · · · , ω(n−1)t}
    ैͬͯɺΑΓҰൠʹɺ
    ∀t ∈ G0
    ; E = F(ωt)
    ͕੒ཱ͢Δɻ
    ͜͜Ͱɺω ͷ F ্ͷ࠷খଟ߲ࣜΛ p(x) ͱ͢Δͱɺf(ω) = p(ω) = 0 ΑΓɺf(x) Λ p(x) Ͱׂͬͨ༨Γ͸߃
    ౳తʹ 0 ʹͳΓɺf(x) = p(x)g(x) ͕੒ཱ͢Δɻैͬͯɺp(x) ͷ͢΂ͯͷࠜ͸ f(x) ͷࠜͰ΋͋ΓɺA ʹؚ·
    ΕΔɻͨͩ͠ɺωt (t ∈ G0
    ) Ҏ֎͸ɺp(x) ͷࠜʹ͸ͳΓಘͳ͍ɻͳͥͳΒɺͦΕҎ֎ͷݩΛ ωk ͱͨ࣌͠ɺk ͱ
    n ͸ޓ͍ʹૉͰͳ͍͜ͱ͔Βɺ࠷େެ໿਺Λ d ͱͯ͠ɺk = a1
    d, n = a2
    d (1 ≤ a2
    < n) ͱͳΓɺa2
    k = a1
    n
    ΑΓɺωa2k = (ωn)a1 = 1 ͱͳΔɻͭ·Γɺωk ͸ xa2 − 1 ͷࠜͰ͋ΓɺԾʹ p(x) ͷࠜͰ΋͋Δͱ͢Δͱɺ
    p(x) = (xa2 − 1)g(x)
    ΋͘͠͸ɺ
    (xa2 − 1) = p(x)g(x)
    ͷͲͪΒ͔͕੒ཱ͢Δ͕ɺલऀ͸ p(x) ͕ن໿Ͱ͋Δ͜ͱʹໃ६ͯ͠ɺޙऀ͸ɺx = ω Λ୅ೖͨ࣌͠ʹɺa2
    < n
    ΑΓ ωa2 = 1 ͱͳΓɺp(ω) = 0 ʹໃ६͢Δɻैͬͯɺp(x) ͷ૬ҧͳΔࠜͷ͢΂ͯ͸
    {ωt | t ∈ G′
    0
    ⊆ G0
    }
    ͱ͍͏ܗͰද͞ΕΔɻ
    ͜͜Ͱɺࣸ૾ͷू߹ {σt
    | t ∈ G′
    0
    } Λ
    x =
    n−1
    i=0
    ai
    ωi ∈ F(ω) (a0
    , · · · , ak
    ∈ F)
    ʹରͯ͠ɺ
    σt
    (x) =
    k
    i=0
    ai
    (ωt)i
    Ͱఆٛ͢Δͱɺ͜ΕΒ͸ɺAut(F(ω)/F) ͷ෦෼ू߹ͱͳΓɺ
    {σt
    | t ∈ G′
    0
    } ⊆ Aut(F(ω)/F) (53)
    ͕੒ཱ͢ΔɻͳͥͳΒɺ্هͷఆٛΛ F(ω) ͔Β F(ωt) ΁ͷࣸ૾ͱߟ͑Δͱɺω ͱ ωt ͸ಉҰͷ࠷খଟ߲ࣜ
    p(x) Λ࣋ͭͷͰɺF(ω) ∼
    = F[x]/p(x)ɺ͓ΑͼɺF(ωt) ∼
    = F[x]/p(x) ͱ͍͏ಉܕؔ܎͕͋ΓɺF[x]/p(x) ͷݩ
    Ͱߟ͑Δͱɺ
    σt
    : F(ω) ∼
    = F[x]/p(x) −→ F[x]/p(x) ∼
    = F(ωt)
    g(x)/p(x) −→ g(x)/p(x)
    42

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  43. ͱ͍͏߃౳ࣸ૾ʹա͗ͣɺF(ω) ͔Β F(ωt) ΁ͷಉܕࣸ૾Ͱ͋Δ͜ͱ͕෼͔Δɻ͞Βʹɺࠓͷ৔߹ɺF(ω) =
    F(ωt) = E ͳͷͰɺ͜Ε͸݁ہɺF(ω) ্ͷʢF ͷݩΛಈ͔͞ͳ͍ʣࣗݾಉܕࣸ૾ͱͳΔɻ
    ҰํɺF(ω) ͸ p(x) ͷ F ্ͷ࠷খ෼ղମͳͷͰɺఆཧ 15 (a) ΑΓ F(ω)/F ͸ΨϩΞ֦େͰ͋Γɺܥ 2ɺ
    ͓Αͼɺఆཧ 2 (b) ΑΓɺ
    |Aut(F(ω)/F)| = [F(ω) : F] = |G′
    0
    | (54)
    ͱͳΔɻ࠷ޙͷ౳߸͸ɺ࠷খଟ߲ࣜ p(x) ͷ࣍਺͸ͦͷࠜͷ਺ |G′
    0
    | ʹ౳͍͜͠ͱΛ༻͍ͨɻैͬͯɺ(53) (54)
    ΑΓɺ
    Aut(F(ω)/F) = {σt
    | t ∈ G′}
    ͕੒ཱ͢Δɻ࠷ޙʹ σt
    ͸ࣸ૾ͷ߹੒ʹؔͯ͠ɺ
    σt1
    ◦ σt2
    = σt3
    (t3
    = t1
    t2
    (mod n))
    ͱ͍͏ؔ܎Λຬͨ͢ͷͰɺG ͷ෦෼܈ʹಉܕͰ͋Δ͜ͱ͕෼͔Δɻ ˙
    ͜ͷิ୊Λ΋ͱʹͯ͠ɺ΂͖֦ࠜେɺ͢ͳΘͪɺ΂͖ࠜΛ෇Ճͯ͠ಘΒΕΔ֦େͱՄղ܈ͷؔ܎Λࣔ͢ɺ࣍
    ͷఆཧ͕ಘΒΕ·͢ɻ
    ఆཧ 17 ମ F ্ͷଟ߲ࣜ f(x) = xn − a ͷ෼ղମΛ E ͱ͢Δ࣌ɺAut(E/F) ͸Մղ܈ͱͳΔɻ·ͨɺ
    αn = a Λຬͨ͢ α ∈ E Λ༻͍ͯɺE = F(α, ω) ͕੒ཱ͢Δɻ͜͜ʹɺω ͸ 1 ͷݪ࢝ n ৐ࠜͰ͋Γɺ
    {α, αω, · · · , αωn−1} ⊂ E
    ͕ f(x) ͷ૬ҧͳΔ n ݸͷࠜͱͳΔɻ
    ʢূ໌ʣ
    ิ୊ 9 ͷ ω ∈ E Λ༻͍ͯɺମͷ֦େͷྻ E ⊃ F(ω) ⊃ F Λߏ੒্ͨ͠Ͱɺ࣍ͷࣗݾಉܕ܈ͷྻ͕Մղ܈
    ͷ৚݅Λຬͨ͢͜ͱΛূ໌͢Δɻ
    Aut(E/F) ⊃ Aut(E/F(ω)) ⊃ {1}
    ·ͣɺӈଆͷϖΞʹΑΔ৒༨܈
    Aut(E/F(ω))/{1}
    ͢ͳΘͪɺAut(E/F(ω)) ͕Ξʔϕϧ܈Ͱ͋Δ͜ͱΛࣔ͢ɻE ͸ xn − a ͷ෼ղମͳͷͰɺαn = a Λຬͨ͢
    α ∈ E ͕ଘࡏ͢Δɻ͜ͷ࣌ɺ{α, αω, αω2, · · · , αωn−1} ͸ɺn ݸͷ૬ҧͳΔݩͰɺ͢΂ͯ f(x) = xn − a ͷ
    ࠜʹͳ͍ͬͯΔɻͭ·ΓɺF(α, ω) ͸ f(x) ͷ෼ղମͰ͋Γɺ෼ղମͷҰҙੑʢఆཧ 16ʣΑΓɺE = F(α, ω)
    ͕੒ΓཱͭɻैͬͯɺAut (E/F(ω)) ͷݩ͸ɺα ʹର͢Δ࡞༻ͷΈͰఆٛ͞ΕΔɻ
    ͜ͷ࣌ɺ೚ҙͷ τ, σ ∈ Aut(E/F(ω)) ʹ͍ͭͯɺα ʹର͢Δ࡞༻͕Մ׵ʹͳΔ͜ͱ͕ɺ࣍ͷٞ࿦͔Β෼͔
    Δɻ·ͣɺf(τ(α)) = τ(f(α)) = 0 ΑΓɺ(τ(α))n = a = αn ͱͳΔɻैͬͯɺ
    τ(α)
    α
    n
    = 1 Ͱ͋Δ͜ͱ͔
    Βɺ
    τ(α)
    α
    ͸ 1 ͷ n ৐ࠜͷ 1 ͭͰ͋Γɺ
    τ(α)
    α
    ∈ {1, ω, · · · , ωn−1} ⊂ F(ω)
    43

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  44. ͕੒ΓཱͭͷͰɺ
    τ(α)
    α
    ͸ σ ͰෆมͱͳΔɻ͜ΕΑΓɺ࣍ͷܭࢉ͕੒Γཱͭɻ
    σ ◦ τ(α) {σ(α)}−1 = σ (τ(α)) σ
    1
    α
    = σ
    τ(α)
    α
    =
    τ(α)
    α
    Αͬͯɺ
    σ ◦ τ(α) =
    σ(α)τ(α)
    α
    Ͱ͋Γɺ্ࣜͷӈล͸ σ ͱ τ ʹ͍ͭͯରশͳͷͰɺσ ◦ τ(α) = τ ◦ σ(α) ͕੒ཱ͢Δɻ͜ΕͰɺAut(E/F(ω))
    ͸Ξʔϕϧ܈Ͱ͋Δ͜ͱ͕ࣔ͞Εͨɻ
    ࣍ʹɺࠨଆͷϖΞʹΑΔ৒༨܈
    Aut(E/F)/Aut(E/F(ω))
    ͕Ξʔϕϧ܈ʹͳΔ͜ͱΛࣔ͢ɻମͷ֦େͷྻ E ⊃ F(ω) ⊃ F ʹ͓͍ͯɺF(ω) ͸ɺxn − 1 ∈ F[x] ͷ෼ղମ
    ͳͷͰɺఆཧ 15 (a) ΑΓɺF(ω)/F ͸ΨϩΞ֦େͰ͋Δɻ͕ͨͬͯ͠ɺఆཧ 10 (a) ΑΓɺAut(E/F(ω)) ͸
    Aut(E/F) ͷਖ਼ن෦෼܈Ͱ͋Γɺ͔֬ʹ্هͷ৒༨܈͕ఆٛͰ͖Δɻ͞Βʹɺఆཧ 11 ΑΓɺ࣍ͷಉܕ͕੒ཱ
    ͢Δɻ
    Aut(E/F)/Aut(E/F(ω)) ∼
    = Aut(F(ω)/F)
    ࠷ޙʹɺิ୊ 9 ΑΓ Aut(F(ω)/F) ͸Ξʔϕϧ܈ͳͷͰɺ͜ΕͰఆཧ͕ূ໌͞Εͨɻ ˙
    7.2 ୅਺తʹՄղͳଟ߲ࣜ
    ͜͜Ͱ͸ɺ͍Α͍Αɺ୅਺ํఔࣜͷղͷެ͕ࣜଘࡏ͢Δ͔Ͳ͏͔ͷٞ࿦Λߦ͍·͢ɻͦ͜Ͱɺ·ͣ͸ɺ
    ʮ୅਺
    ํఔࣜͷղͷެࣜʯͱ͸Կ͔ΛվΊͯߟ͑௚ͯ͠Έ·͢ɻͨͱ͑͹ɺఆཧ 14 ʹΑΓɺ೚ҙͷଟ߲ࣜ f(x) ʹ
    ରͯ͠ɺ͜ΕΛҼ਺෼ղՄೳʹ͢Δ࠷খ෼ղମΛఆٛ͢Δ͜ͱ͕Ͱ͖·͢ɻ͔͠͠ͳ͕Βɺ͜Ε͸ɺ࠷খଟ߲
    ࣜ p(x) Λ༻͍ͨ৒༨ମ F[x]/p(x) Λܗࣜతʹߏ੒͍ͯ͠Δ͚ͩͰ͋Γɺ۩ମతʹ f(x) = 0 Λຬͨ͢ x Λٻ
    ΊΔखଓ͖Λ༩͍͑ͯΔΘ͚Ͱ͸͋Γ·ͤΜɻ
    ͦΕͰ͸ɺղͷެ͕ࣜ஌ΒΕ͍ͯΔ 2 ࣍ҎԼͷଟ߲ࣜͷ৔߹ɺղͷެࣜΛ༻͍ͯղ x Λܭࢉ͢Δखଓ͖͸ɺ
    ۩ମతʹͲͷΑ͏ʹͳ͍ͬͯΔͰ͠ΐ͏͔ʁ ͨͱ͑͹ɺ1 ࣍ํఔࣜ
    ax + b = 0 (a, b ∈ F)
    ͷղͷެࣜ͸ɺ໌Β͔ʹ
    x = −
    b
    a
    Ͱ༩͑ΒΕ·͢ɻ͜ͷ৔߹ɺղ x ͸ମ F ͷ࢛ଇԋࢉͷΈͰදݱ͞Ε͓ͯΓɺ৽ͨͳ֦େମΛಋೖ͢Δඞཁ͸
    ͋Γ·ͤΜɻҰํɺ2 ࣍ํఔࣜ
    ax2 + bx + c = 0 (a, b, c ∈ F)
    ͷղͷެࣜ͸ɺ
    x =
    −b ±

    b2 − 4ac
    2a
    Ͱ༩͑ΒΕ·͢ɻ͜ͷ৔߹ɺ࣮ࡍʹ͜ͷެࣜΛ༻͍ͯ x Λܭࢉ͢ΔखॱΛߟ͑Δͱɺ͸͡ΊʹɺF ͷ࢛ଇԋ
    ࢉʹΑͬͯ b2 − 4ac Λܭࢉͯ͠ɺͦͷޙɺଟ߲ࣜ x2 − (b2 − 4ac) ͷࠜ α Λಋೖͯ͠ɺମΛ F(α) ʹ֦େ͠
    44

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  45. ·͢ɻͦͷޙɺF(α) ͷ࢛ଇԋࢉʹΑͬͯɺ
    x =
    −b ± α
    2a
    ͱͯ͠ɺղ x ͷ஋͕ಘΒΕ·͢ɻ͜͜Ͱɺα ͸ɺxn − a ͱ͍͏ܗࣜͷଟ߲ࣜͷղɺ͢ͳΘͪɺ΂͖ࠜͰ͋Δ
    ͜ͱ͕෼͔Γ·͢ɻͭ·Γɺղͷެࣜͱ͸ɺग़ൃ఺ͱͳΔମ F ʹରͯ͠ɺମͷ࢛ଇԋࢉɺ͓Αͼɺ΂͖ࠜͰ
    දݱ͞ΕΔݩΛ௥Ճͯ͠ମΛ֦େ͢Δͱ͍͏ૢ࡞Λ܁Γฦ͢͜ͱͰɺx Λܭࢉ͢Δखଓ͖Λ༩͑Δ΋ͷͱཧղ
    ͢Δ͜ͱ͕Ͱ͖·͢ɻ͜ͷཧղͷԼʹɺղͷެ͕ࣜଘࡏ͢Δଟ߲ࣜɺ͢ͳΘͪɺ୅਺తʹՄղͳଟ߲ࣜΛ࣍ͷ
    Α͏ʹఆٛ͠·͢ɻ
    ఆٛ 2 ମ F ্ͷଟ߲ࣜ f(x) ͷ࠷খ෼ղମΛ E ͱ͢Δ࣌ɺ༗ݶݸͷ֦େମͷྻ
    Fm
    ⊃ · · · ⊃ F1
    ⊃ F0
    = F
    ͕ଘࡏͯ͠ɺFm
    ⊇ E Ͱ͋ΓɺͦΕͧΕͷ֦େ͕΂͖֦ࠜେͰ͋Δɺͭ·Γɺఆཧ 17 ʹΑΓɺ
    Fk
    = Fk−1
    (α, ωn
    ) (k = 1, · · · , m)
    ͕੒Γཱͭͱ͢Δɻ͜͜Ͱɺα ͸ɺద౰ͳࣗવ਺ n ≥ 2 ͱ a ∈ Fk−1
    ʹର͢Δଟ߲ࣜ xn − a ͷࠜͰɺωn
    ͸ɺ
    1 ͷݪ࢝ n ৐ࠜͱ͢Δɻ
    ʢͨͩ͠ɺn = 2 ͷ৔߹͸ɺωn
    = −1 ͳͷͰɺ࣮ࡍʹ͸ ωn
    Λ෇༩͢Δඞཁ͸ͳ͍ɻ
    ʣ
    ͜ͷ࣌ɺଟ߲ࣜ f(x) ͸ F ্Ͱ୅਺తʹՄղͰ͋Δͱ͍͏ɻ
    ఆٛ 2 ͷ৚͕݅ຬͨ͞ΕΔ৔߹ɺͦΕͧΕͷ֦େྻ F0
    , F1
    , · · · ʹ͓͚Δ࢛ଇԋࢉɺ͓Αͼɺ΂͖ࠜΛٻΊ
    Δʢͭ·Γɺ͋Δ஋ a ʹରͯ͠ n

    a ΛٻΊΔʣͱ͍͏ૢ࡞ͷ܁Γฦ͠ʹΑͬͯɺղ x Λܭࢉ͢Δ͜ͱ͕Մೳ
    ʹͳΓ·͢ɻٯʹ͜ͷ৚͕݅ຬͨ͞Εͳ͚Ε͹ɺͦͷΑ͏ͳҰൠతͳૢ࡞͸ଘࡏͤͣɺ͍ΘΏΔʮղͷެࣜʯ
    ͸࡞Εͳ͍͜ͱʹͳΓ·͢ɻ
    ͦͯ͠ɺ͜ͷ৚݅͸ɺ࠷খ෼ղମΛ E ͱͯ͠ɺ֦େ E/F ͷΨϩΞ܈͕Մղ܈Ͱ͋Δ͜ͱͱಉ஋ʹͳΔͱ
    ͍͏ͷ͕ɺ࠷ऴతͳ݁࿦Ͱ͢ɻ͜ͷ݁ՌΛॱΛ௥͍͖ͬͯࣔͯ͠·͢ɻ·ͣ͸४උͱͯ͠ɺิ୊Λ 2 ͭࣔ͠
    ·͢ɻ
    ิ୊ 10 ω Λ 1 ͷݪ࢝ n ৐ࠜͱ͢Δ࣌ɺΨϩΞ֦େ E/F(ω) ʹରͯ͠ɺAut(E/F(ω)) ͕८ճ܈Ͱɺ
    |Aut(E/F(ω))| = n Ͱ͋ͬͨͱ͢Δɻ͜ͷ࣌ɺE/F(ω) ͸ n ࣍ͷ΂͖֦ࠜେͰ͋Δɻ͢ͳΘͪɺ
    ∃α; αn ∈ F(ω) s.t. E = F(α, ω)
    ͕੒Γཱͭɻ
    ʢূ໌ʣ
    E/F(ω) ͕ΨϩΞ֦େͰ͋Δ͜ͱ͔Βɺఆཧ 6 ʹΑΓɺ
    EAut(E/F (ω)) = F(ω) (55)
    ͕੒Γཱͭɻ·ͨɺE Λ F(ω) ্ͷϕΫτϧۭؒͱݟͳͨ࣌͠ɺܥ 2 ΑΓɺϕΫτϧۭؒͷ࣍ݩ͸ɺ
    [E : F(ω)] = |Aut(E/F(ω))| = n
    ͱܾ·Δɻ͞Βʹɺ८ճ܈ Aut(E/F(ω)) ͷੜ੒ݩΛ σ ͱ͢Δͱɺ|Aut(E/F(ω))| = n Ͱ͋Δ͜ͱ͔Βɺ
    Aut(E/F(ω)) = {1, σ, σ2, · · · , σn−1} (σn = 1) (56)
    45

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  46. ͱͳΔɻ
    ͜͜Ͱɺσ Λલड़ͷϕΫτϧۭؒʹ͓͚Δઢܗࣸ૾ͱΈͳͯ͠ɺݻ༗ํఔࣜ
    σ(α) = λα (λ ∈ F(ω), α ∈ E)
    Λߟ͑Δͱɺσn = 1 ΑΓɺݻ༗ϕΫτϧ α ʹରͯ͠ α = λnα ͕੒ཱ͢Δɻͭ·Γɺݻ༗஋ λ ͸͢΂ͯ 1 ͷ
    n ৐ࠜͰ͋Γɺ͜ΕΑΓɺn ݸͷݻ༗஋͸ {1, ω, · · · , ωn−1} ʹܾ·Δɻͦ͜Ͱɺಛʹɺݻ༗஋ ω ʹରԠ͢Δ
    ݻ༗ϕΫτϧΛ α ͱ͢Δͱɺσ(α) = ωα Λ༻͍ͯɺ
    σ(αn) = {σ(α)}n = ωnαn = αn
    ͕੒ཱ͢Δɻͭ·Γɺ
    αn ͸ σ ∈ Aut(E/F(ω)) Ͱಈ͔ͳ͍ݩͰ͋Γɺ
    (55) ΑΓɺ
    αn ∈ F(ω) ͱͳΔɻैͬͯɺ
    xn − αn ∈ F(ω)[x] (57)
    Ͱ͋Γɺ{α, αω, · · · , αωn−1} ⊂ E ͸ɺ(57) ͷ n ݸͷ૬ҧͳΔࠜͰ͋Δɻ͜ΕΑΓɺF(α, ω) ͸ (57) ͷ࠷খ
    ෼ղମͰ͋Δ͜ͱʹͳΔɻ
    ·ͨɺAut(E/F(ω)) ⊇ Aut(E/F(α, ω)) ʹ஫ҙ͢ΔͱɺAut(E/F(α, ω)) ͷݩ͸ɺ(56) ʹؚ·ΕΔ͜ͱʹ
    ͳΔ͕ɺ
    σi(α) = ωiα = α (i = 1, · · · , n − 1)
    ΑΓɺα ∈ F(α, ω) Λݻఆ͢Δͷ͸߃౳ࣸ૾ 1 ͷΈʹͳΔɻͭ·Γɺ
    Aut(E/F(α, ω)) = {1}
    ͱͳΔɻ͜ͷ࣌ɺఆཧ 8 ΑΓ֦େ E/F(α, ω) ͸ΨϩΞ֦େͳͷͰɺܥ 2 Λ༻͍ͯɺ
    [E : F(α, ω)] = |Aut(E/F(α, ω))| = 1
    ͕ಘΒΕΔɻैͬͯɺ
    E ͷ F(α, ω) ্ͷجఈΛ e ͱͯ͠ɺ
    ೚ҙͷ x ∈ E ͸ɺ
    e ͱ F(α, ω) ͷݩͷੵͰදݱ͞Ε
    Δ͜ͱʹͳΔɻಛʹ x = 1 ͷ৔߹Λߟ͑Δͱɺe ͸ F(α, ω) ͷݩʹର͢ΔੵͷٯݩͰ͋Γɺ݁ہ e ∈ F(α, ω)
    ͱͳΔ͜ͱ͕෼͔Γɺ
    E = F(α, ω)
    ͕੒ΓཱͭɻैͬͯɺE ͸ (57) ͷ࠷খ෼ղମͰ͋Γɺ֦େ E/F(ω) ͸΂͖֦ࠜେͰ͋Δɻ ˙
    ิ୊ 11 ༗ݶ࣍ݩͷΨϩΞ֦େ E/F ʹ͍ͭͯɺΨϩΞ܈ G = Aut(E/F) ͕Ξʔϕϧ܈Ͱ͋Δ࣌ɺத֦ؒ
    େͷྻ
    E = A0
    ⊃ A1
    ⊃ · · · ⊃ Al
    = F
    ͱɺͦΕʹରԠ͢ΔΨϩΞ܈ͷ෦෼܈ͷྻ
    {1} = Aut(E/A0
    ) ⊂ Aut(E/A1
    ) ⊂ · · · ⊂ Aut(E/Al
    ) = Aut(E/F)
    ͕ଘࡏͯ͠ɺྡΓ߹͏܈ͷ৒༨܈͸͢΂ͯ८ճ܈ͱͳΔɻ
    46

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  47. ʢূ໌ʣ
    E/F ͸ΨϩΞ֦େͳͷͰɺܥ 2 ΑΓɺ
    |G| = [E : F] < ∞
    ͕੒ΓཱͪɺG ͸༗ݶΞʔϕϧ܈ͱͳΔɻैͬͯɺΞʔϕϧ܈ͷجຊఆཧʹΑΓɺG ͸༗ݶݸͷ८ճ܈ͷ௚
    ੵʹ෼ղ͞ΕΔɻ
    G ∼
    = C1
    × · · · × Cl
    ͜͜Ͱɺ
    Gk
    = C1
    × · · · × Ck
    × {1} × · · · × {1}
    l−k ݸ
    (k = 1, · · · , l)
    ͱͯ͠ɺதؒମͷྻ
    E = A0
    ⊃ A1
    ⊃ · · · ⊃ Al
    = F
    Λߏ੒͢Δɻ͜͜ʹɺ
    Ak
    = EGk (k = 1, · · · , l)
    Ͱ͋ΓɺE/F ͕ΨϩΞ֦େͰ͋Δ͜ͱ͔ΒɺAl
    = EG = F ͕੒Γཱͭ͜ͱΛ༻͍͍ͯΔɻ͜ͷ࣌ɺఆཧ 9
    ΑΓɺ͜ΕʹରԠ͢Δ෦෼܈ͷྻ͕ಘΒΕΔɻ
    {1} = Aut(E/A0
    ) ⊂ Aut(E/A1
    ) ⊂ · · · ⊂ Aut(E/Al
    ) = Aut(E/F) (58)
    ͜͜Ͱɺఆཧ 5 ΑΓɺ
    Aut(E/Ak
    ) = Aut(E/EGk ) = Gk

    = C1
    × · · · × Ck
    ͱͳΔͷͰɺ
    Aut(E/Ak
    )/Aut(E/Ak−1
    ) ∼
    = Ck
    (k = 1, · · · , l)
    ͕੒ཱ͢Δɻैͬͯɺ(58) ͕ٻΊΔ෦෼܈ͷྻͱͳΔɻ ˙
    ͦΕͰ͸ɺ͜͜Ͱɺຊ୊ͷఆཧΛূ໌͢Δɻ
    ఆཧ 18 F ্ͷଟ߲ࣜ f(x) ͷ࠷খ෼ղମΛ E ͱ͢Δ࣌ɺf(x) ͕୅਺తʹՄղͰ͋Δ͜ͱ͸ɺAut(E/F)
    ͕Մղ܈Ͱ͋Δ͜ͱͱಉ஋Ͱ͋Δɻ
    ʢূ໌ʣ
    લఏͱͯ͠ɺE ͸ f(x) ͷ F ্ͷ࠷খ෼ղମͳͷͰɺఆཧ 15 (a) ΑΓɺE/F ͸ΨϩΞ֦େͰ͋Δɻ
    ɾ୅਺తʹՄղ ⇒ Մղ܈
    ࠓɺf(x) ͸୅਺తʹՄղͩͱ͢Δͱɺf(x) ͷ෼ղମ E ʹରͯ͠ɺՄղੑͷఆٛΛຬͨ͢ɺ΂͖֦ࠜେͷྻ
    Fm
    (⊃ E) ⊃ Fm−1
    ⊃ · · · ⊃ F1
    ⊃ F0
    = F
    ͕ଘࡏ͢Δɻ͜ͷ࣌ɺ೚ҙͷ r (r = 1, · · · , m) ʹ͍ͭͯɺFr
    ͸࣍ͷ৚݅Λຬͨ͢͜ͱΛ਺ֶతؼೲ๏Ͱূ໌
    ͢Δɻ
    47

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  48. ɾAut(Fr
    /F) ͸Մղ܈
    ɾFr
    /F ͸ΨϩΞ֦େ
    ͜Ε͕ূ໌Ͱ͖ͨͱ͢ΔͱɺE ͕ΨϩΞ֦େ Fm
    /F ͷதؒମ Fm
    ⊃ E ⊃ F Ͱɺ͔ͭɺE/F ͕ΨϩΞ֦େ
    Ͱ͋Δ͜ͱ͔Βɺఆཧ 11 ΑΓɺ
    Aut(Fm
    /F)/Aut(Fm
    /E) ∼
    = Aut(E/F)
    ͕੒ΓཱͭɻैͬͯɺAut(Fm
    /F) ͕Մղ܈Ͱ͋Δ͜ͱ͔Βɺఆཧ 12 ʹΑΓɺAut(E/F) ͕Մղ܈Ͱ͋Δ͜
    ͱ͕ূ໌͞ΕΔɻ
    ·ͣɺr = 1 ͷ࣌Λߟ͑ΔͱɺF1
    /F ͸΂͖֦ࠜେͳͷͰɺఆཧ 17 ΑΓ Aut(F1
    /F) ͸Մղ܈ʹͳΔɻಉ͡
    ͘ɺఆཧ 15 (a) ΑΓ F1
    /F ͸ΨϩΞ֦େʹͳΔɻ࣍ʹɺr − 1 ·Ͱ੒ཱ͢ΔͱԾఆͯ͠ɺr ≥ 2 ͷ৔߹Λߟ
    ͑Δɻ
    ࠓɺFr
    /Fr−1
    ͸ xn − a ʹΑΔ΂͖֦ࠜେͳͷͰɺఆཧ 17 ΑΓɺFr
    = Fr−1
    (α, ω) ͱͳΔ α, ω ͕ଘࡏ͢
    Δɻ͜͜ʹɺαn = aɺ͓Αͼɺω ͸ 1 ͷݪ࢝ n ৐ࠜͰ͋Δɻ·ͨɺFr−1
    /F ͕ΨϩΞ֦େͰ͋ΔͷͰɺఆཧ
    15 (a) ΑΓɺFr−1
    ͸͋Δଟ߲ࣜ g(x) ͷ࠷খ෼ղମͰ͋ΓɺFr
    = Fr−1
    (α, ω) ͸ g(x)(xn − a) ͷ࠷খ෼ղମ
    ͱͳΔɻैͬͯɺ࠶౓ɺఆཧ 15 (a) ΑΓɺFr
    /F ͕ΨϩΞ֦େͱͳΔɻͭ·Γɺ֦େͷྻ Fr
    ⊃ Fr−1
    ⊃ F ʹ
    ͓͍ͯɺFr
    /F ͱ Fr−1
    /F ͕ڞʹΨϩΞ֦େͰ͋Γɺఆཧ 11 ΑΓɺ͕࣍੒ཱ͢Δɻ
    Aut(Fr
    /F)/Aut(Fr
    /Fr−1
    ) ∼
    = Aut(Fr−1
    /F)
    ͜͜Ͱɺఆཧ 17 ΑΓ Aut(Fr
    /Fr−1
    ) ͸Մղ܈Ͱɺ͞Βʹ Aut(Fr−1
    /F) ΋Մղ܈Ͱ͋Δ͜ͱ͔Βɺఆཧ 13
    ʹΑΓɺAut(Fr
    /F) ͸Մղ܈ͱͳΔɻ͜ΕͰ Fr
    ͸ূ໌͢Δ΂͖ੑ࣭Λ͢΂͍ͯ࣋ͬͯΔ͜ͱ͕෼͔ͬͨɻ
    ɾՄղ܈ ⇒ ୅਺తʹՄղ
    Aut(E/F) ͕Մղ܈Ͱ͋Δ͜ͱ͔ΒɺAut(E/F) ͷ෦෼܈ͷྻ͕ଘࡏ͢Δ͕ɺఆཧ 9 ΑΓɺ͜Ε͸ɺE ⊃ F
    ͷதؒମͷྻʹରԠ͢Δɻ۩ମతʹ͸ɺதؒମͷྻΛ
    E = Fm
    ⊃ Fm−1
    ⊃ · · · ⊃ F0
    = F (59)
    ͱͯ͠ɺ͜ΕʹରԠ͢Δࣗݾಉܕࣸ૾ͷ෦෼܈ͷྻ
    {1} = Aut(E/Fm
    ) ⊂ Aut(E/Fm−1
    ) ⊂ · · · ⊂ Aut(E/F0
    ) = Aut(E/F) (60)
    ͕Մղ܈ͷྻΛߏ੒͢Δɻ͜ͷ࣌ɺྡΓ߹͏܈͸ਖ਼ن෦෼܈Ͱ͋ΓɺͦΕΒͷ৒༨܈͸Ξʔϕϧ܈ʹͳΔɻ·
    ͨɺ೚ҙͷ k ʹ͍ͭͯɺE/Fk
    ͸ɺΨϩΞ֦େ E/F ͷத֦ؒେͳͷͰΨϩΞ֦େʹͳΔɻ
    ͜ͷ࣌ɺ೚ҙͷྡΓ߹͏֦େମ E ⊃ Fk+1
    ⊃ Fk
    ʹ͓͍ͯɺE/Fk
    ͸ΨϩΞ֦େͰɺ͞ΒʹɺAut(E/Fk+1
    )
    ͕ Aut(E/Fk
    ) ͷਖ਼ن෦෼܈Ͱ͋Δ͜ͱ͔Βɺఆཧ 10 (a) ʹΑΓɺFk
    /Fk+1
    ͸ΨϩΞ֦େͱͳΔɻΑͬͯɺ
    ఆཧ 11 ʹΑΓɺ࣍ͷಉܕ͕੒ཱ͢Δɻ
    Aut(E/Fk+1
    )/Aut(E/Fk
    ) ∼
    = Aut(Fk
    /Fk+1
    ) (61)
    (61) ͷࠨล͸ (60) ͷྡΓ߹͏܈ͷ৒༨܈ͳͷͰΞʔϕϧ܈Ͱ͋ΓɺAut(Fk
    /Fk+1
    ) ͸Ξʔϕϧ܈ͱͳΔɻ͠
    ͕ͨͬͯɺิ୊ 11 Λ༻͍ͯɺ֦େ Fk
    /Fk+1
    Λ͞Βʹதؒମʹ෼ׂ͢Δ͜ͱͰɺ(59) (60) ͸ɺྡΓ߹͏܈ͷ
    ৒༨܈͕͢΂ͯ८ճ܈ͱͳΔྻʹ෼ׂ͢Δ͜ͱ͕Ͱ͖Δɻ͜ΕҎ߱͸ɺ(59) (60) ΛͦͷΑ͏ʹ෼ׂͨ͠΋ͷ
    ͱΈͳͯٞ͠࿦ΛਐΊΔɻͭ·Γɺ(61) ʹ͓͍ͯɺAut(Fk
    /Fk+1
    ) ͸८ճ܈Ͱ͋Δͱ͢Δɻ
    48

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  49. ͜ͷલఏͷԼʹɺ(59) ͷ֦େྻΛ΂͖֦ࠜେͷྻʹؼೲతʹ֦ு͍ͯ͘͠ɻ·ͣɺΨϩΞ֦େ F1
    /F0
    ʹ͓
    ͍ͯɺ|Aut(F1
    /F0
    )| = n1
    ͱͯ͠ɺ1 ͷݪ࢝ n1
    ৐ࠜ ω1
    Λ෇༩֦ͨ͠େ F1
    (ω1
    )/F0
    (ω1
    ) Λߏ੒͢Δɻ͜ͷ
    ࣌ɺAut(F1
    (ω1
    )/F0
    (ω1
    )) ͷཁૉΛߟ͑Δͱɺ͜Ε͸ɺF1
    ͷݩʹର͢Δࣸ૾ͱ ω1
    ʹର͢Δࣸ૾Ͱܾఆ͞ΕΔ
    ͕ɺω1
    ͸ಈ͔͞ͳ͍͜ͱ͔Βɺ݁ہɺ
    Aut(F1
    (ω1
    )/F0
    (ω1
    )) ∼
    = Aut(F1
    /F0
    )
    ͱͳΔɻैͬͯɺAut(F1
    /F0
    ) ͕८ճ܈Ͱ͋Δ͜ͱ͔ΒɺAut(F1
    (ω1
    )/F0
    (ω1
    )) ΋८ճ܈ͱͳΓɺ͞Βʹɺ
    |Aut(F1
    (ω1
    )/F0
    (ω1
    ))| = n1
    ͱͳΔ͜ͱ͔Βɺ֦େ F1
    (ω1
    )/F0
    (ω1
    ) ʹରͯ͠ิ୊ 10 ͕ద༻Ͱ͖ͯɺF1
    (ω1
    )/F0
    (ω1
    ) ͸΂͖֦ࠜେͱͳΔɻ
    ͞ΒʹɺF2
    /F1
    ʹ͓͍ͯɺ|Aut(F2
    /F1
    )| = n2
    ͱͯ͠ɺω1
    ɺ͓Αͼɺ1 ͷݪ࢝ n2
    ৐ࠜ ω2
    Λ෇༩֦ͨ͠େ
    F2
    (ω1
    , ω2
    )/F1
    (ω1
    , ω2
    ) Λߏ੒͢Δɻઌ΄Ͳͱಉ༷ʹɺ
    Aut(F2
    (ω1
    , ω2
    )/F1
    (ω1
    , ω2
    )) ∼
    = Aut(F2
    /F1
    )
    Ͱ͋Δ͜ͱ͔Βɺิ୊ 10 ΑΓ Aut(F2
    (ω1
    , ω2
    )/F0
    (ω1
    , ω2
    )) ͸΂͖֦ࠜେͱͳΔɻ͜ͷஈ֊Ͱɺ࣍ͷ֦େྻΛ
    ߟ͑Δͱɺ͜Ε͸΂͖֦ࠜେͷྻʹͳΔ͜ͱ͕Θ͔Δɻ
    ʢF1
    (ω1
    , ω2
    )/F1
    (ω1
    ) ͸΂͖ࠜ ω2
    Λ෇༩֦ͨ͠େͳͷ
    Ͱɺ΂͖֦ࠜେͰ͋Δɻ
    ʣ
    F2
    (ω1
    , ω2
    ) ⊃ F1
    (ω1
    , ω2
    ) ⊃ F1
    (ω1
    ) ⊃ F0
    (ω1
    ) ⊃ F0
    ͜ͷखଓ͖Λ܁Γฦ͢͜ͱͰɺ࠷ऴతʹ E(ω1
    , · · · , ωm
    ) ʹ͍ͨΔɺ΂͖֦ࠜେͷྻ͕ߏ੒Ͱ͖ͯɺ
    E(ω1
    , · · · , ωm
    ) ⊃ E Ͱ͋Δ͜ͱ͔Βɺf(x) ͸୅਺తʹՄղͰ͋Δ͜ͱ͕ࣔ͞Εͨɻ ˙
    8 n ࣍ଟ߲ࣜͷՄղੑ
    8.1 n ࣍ํఔࣜͷҰൠղ
    ఆཧ 18 ʹΑΓɺ೚ҙͷଟ߲ࣜ f(x) ʹ͍ͭͯɺ
    ʮ࢛ଇԋࢉͱ΂͖ࠜͷ૊Έ߹ΘͤʹΑͬͯɺͦͷࠜΛܭࢉ͢
    Δखଓ͖͕ଘࡏ͢Δ͔Ͳ͏͔ʯ͕൑ఆͰ͖ΔΑ͏ʹͳΓ·ͨ͠ɻ͜͜Ͱ͸ɺҰൠͷ n ࣍ํఔࣜʹରͯ͠ɺ͜ͷ
    ൑ఆํ๏Λద༻ͯ͠Έ·͢ɻ͜ͷࡍɺํఔࣜͷ܎਺Λߏ੒͢Δମ F ͱ࠷খ෼ղମ E ͷؔ܎Λ஌Δඞཁ͕͋Γ
    ·͢ɻͦ͜Ͱɺ·ͣɺE ্Ͱ 1 ࣍ࣜͷੵʹҼ਺෼ղ͞Εͨଟ߲ࣜΛܗࣜతʹల։ͯ͠Έ·͢ɻ
    n
    i=1
    (x − αi
    ) = xn − s1
    xn−1 + · · · + (−1)nsn
    ͜ͷ࣌ɺల։ͨ͠ޙͷ܎਺ sk
    (k = 1, · · · , n) ͸ɺ࣍ͷجຊରশࣜͰ༩͑ΒΕ·͢ɻ
    sk
    =
    i1<···αi1
    · · · αik
    (62)
    ۩ମతʹ͸ɺ࣍ͷΑ͏ͳܗʹͳΓ·͢ɻ
    s1
    = α1
    + · · · + αn
    s2
    = α1
    α2
    + · · · + αn−1
    αn
    (63)
    .
    .
    .
    sn
    = α1
    · · · αn
    49

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  50. ͜͜Ͱɺ{s1
    , · · · , sn
    } ͸ɺҰൠʹɺ༗ཧ਺ମ Q ΑΓ΋େ͖ͳମͷݩͰ͋Δ΋ͷͱͯ͠ɺ͜ΕΒΛؚΉ࠷খͷ
    ମΛ
    F = Q(s1
    , · · · , sn
    )
    ͱ͠·͢ɻಉ༷ʹɺ{α1
    , · · · , αn
    } ΛؚΉ࠷খͷମΛ
    E = Q(α1
    , · · · , αn
    )
    ͱ͠·͢ɻ(62) ͷؔ܎Λ༻͍Ε͹ɺ{α1
    , · · · , αn
    } ͷ૊Έ߹ΘͤͰ {s1
    , · · · , sn
    } Λදݱ͢Δ͜ͱ͕Ͱ͖Δͷ
    ͰɺE ͸ F ΛؚΉମͰ͋Γɺ
    E ⊇ F
    ͱ͍͏ؔ܎͕੒Γཱͪ·͢ɻΑΓਖ਼֬ʹ͸ɺ͸͡Ίʹɺମ E = Q(α1
    , · · · , αn
    ) Λఆ͓͖ٛͯ͠ɺ͔ͦ͜Βɺ
    (62) Λ௨ͯ͡ɺͦͷ෦෼ମ
    F = Q(α1
    + · · · + αn
    , α1
    α2
    + · · · + αn−1
    αn
    , · · · , α1
    · · · αn
    )
    Λఆ͍ٛͯ͠Δ΋ͷͱߟ͍͑ͯͩ͘͞ɻ͜ͷ࣌ɺ֦େ E/F ͷΨϩΞ܈ Aut(E/F) Λߟ͑Δͱɺ͜ͷཁૉ͸
    Q ͷݩΛಈ͔͢͜ͱ͸Ͱ͖ͣɺ{α1
    , · · · , αn
    } Λޓ͍ʹೖΕସ͑Δૢ࡞ʹݶఆ͞Ε·͢ɻͦͯ͠ɺ(62) ͷؔ܎
    ΑΓɺ͜ΕΒͷஔ׵ૢ࡞ʹΑͬͯɺ{s1
    , · · · , sn
    } ͸มԽ͢Δ͜ͱ͸͋Γ·ͤΜɻͭ·ΓɺAut(E/F) ͸ɺn ݸ
    ͷݩͷஔ׵ૢ࡞ʹରԠͨ͠ɺn ࣍ͷରশ܈ Sn
    ͱಉܕʹͳΓ·͢ɻ
    Aut(E/F) ∼
    = Sn
    ैͬͯɺSn
    ͕Մղ܈Ͱ͋Δ͜ͱ͕ɺ΋ͱͷ n ࣍ํఔ͕ࣜ୅਺తʹՄղͰ͋Δ͜ͱͱಉ஋ʹͳΓ·͢ɻ୅਺త
    ʹՄղͰ͋Ε͹ɺ(63) Ͱఆٛ͞Εͨ {s1
    , · · · , sn
    } Λ༻͍ͯɺମͷ࢛ଇԋࢉͱ΂͖ࠜͷܭࢉΛ૊Έ߹ΘͤΔ͜
    ͱͰɺ{α1
    , · · · , αn
    } Λܭࢉ͢Δ͜ͱ͕ՄೳʹͳΓ·͢ɻͭ·Γɺ(63) ͷؔ܎Λٯղ͖ͨ͠ɺ
    ʮղͷެࣜʯ͕ಘ
    ΒΕΔͱ͍͏Θ͚Ͱ͢ɻ
    Ұൠʹ n ≥ 5 ͷ৔߹ɺରশ܈ Sn
    ͸Մղ܈ʹͳΒͳ͍͜ͱ͕஌ΒΕ͓ͯΓɺ͜ΕʹΑΓɺ5 ࣍Ҏ্ͷଟ߲
    ࣜʹ͍ͭͯɺղͷެࣜʢҰൠղʣΛ࢛ଇԋࢉͱ΂͖ࠜͰදݱ͢Δ͜ͱ͸Ͱ͖ͳ͍ͱ݁࿦෇͚ΒΕ·͢ɻٯʹ
    n ≤ 4 ͷ৔߹ɺରশ܈ Sn
    ͸Մղ܈Ͱ͋Γɺ4 ࣍ҎԼͷଟ߲ࣜ͸ɺ࢛ଇԋࢉͱ΂͖ࠜͰҰൠղΛදݱͰ͖Δ͜
    ͱʹͳΓ·͢ɻͨͩ͠ɺ͜Ε͸ɺ͋͘·ͰͦͷΑ͏ͳૢ࡞͕ଘࡏ͢Δ͜ͱΛ͍ࣔͯ͠Δ͚ͩͰɺ۩ମతͳදݱ
    ํ๏Λ͍ࣔͯ͠ΔΘ͚Ͱ͸͋Γ·ͤΜɻΨϩΞཧ࿦Λ΋ͬͯͯ͠΋ɺ3 ࣍ํఔࣜ΍ 4 ࣍ํఔࣜͷҰൠղͷެࣜ
    Λ໌ࣔతʹࣔ͢͜ͱ͸Ͱ͖ͳ͍ͷͰ͠ΐ͏͔?
    —— ΋ͪΖΜɺͦͷΑ͏ͳ͜ͱ͸͋Γ·ͤΜɻఆཧ 18ɺ͓Αͼɺิ୊ 10 ͷূ໌Ͱ͸ɺՄղ܈ͷྻ͕༩͑
    ΒΕͨ࣌ʹɺରԠ͢Δ΂͖֦ࠜେΛߏ੒͢Δखଓ͖Λ༩͍͑ͯ·͢ɻ͜ͷखଓ͖Λར༻͢Δ͜ͱͰɺղͷެࣜ
    Λಋͨ͘Ίʹඞཁͳ΂͖֦ࠜେΛ۩ମతʹߏ੒͢Δ͜ͱ͕ՄೳʹͳΓ·͢ɻ࣍અͰ͸ɺ2 ࣍ํఔࣜͱ 3 ࣍ํఔ
    ࣜͷ৔߹ʹ͍ͭͯɺ΂͖֦ࠜେͷखଓ͖Λ۩ମతʹߏ੒͢Δ͜ͱͰɺͦΕͧΕͷղͷެࣜΛಋ͖·͢ɻ
    8.2 2 ࣍ํఔࣜͷղͷެࣜ
    લઅͷҰൠతͳखଓ͖ʹैͬͯɺ࣍ͷଟ߲ࣜΛߟ͑·͢ɻ
    f(x) = (x − α1
    )(x − α2
    ) ∈ Q(α1
    , α2
    )[x]
    50

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  51. ͜ΕΛల։͢Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ
    f(x) = x2 − (α1
    + α2
    )x + α1
    α2
    ∈ Q(α1
    + α2
    , α1
    α2
    )[x]
    ͜ͷ࣌ɺମͷ֦େ
    Q(α1
    , α2
    )/Q(α1
    + α2
    , α1
    α2
    )
    ͷΨϩΞ܈͸ɺ2 ࣍ͷରশ܈ S2
    ʢα1
    ͱ α2
    ͷೖΕସ͑ʣͰ͋Γɺ͜Ε͸ɺਖ਼ن෦෼܈ͷྻ
    S2
    ⊃ {1}
    ʹΑΓՄղ܈ͱͳΓ·͢ɻ
    ͪͳΈʹɺମ Q(α1
    + α2
    , α1
    α2
    ) ͷൣғͰͳΜΒ͔ͷܭࢉΛߦ͏ࡍ͸ɺα1
    ͱ α2
    ͸ɺα1
    + α2
    ͱ α1
    α2
    ͱ͍
    ͏૊Έ߹ΘͤͷΈͰѻ͏ඞཁ͕͋Γ·͢ɻ౰વͳ͕Βɺ͜ͷൣғͰ͸ɺα1
    ͱ α2
    ͦͷ΋ͷΛಘΔ͜ͱ͸Ͱ͖
    ·ͤΜɻ
    ͜͜Ͱɺ
    E = Q(α1
    , α2
    )
    F = Q(α1
    + α2
    , α1
    α2
    )
    ͱஔ͘ͱɺ
    [E : F] = |Aut(E/F)| = |S2
    | = 2
    ͱͳΓ·͕͢ɺS2
    ͕८ճ܈Ͱ͋Δ͜ͱ͔Βɺ͜Ε͸ิ୊ 10 Ͱ n = 2 ͱஔ͍ͨ৔߹ͱಉ͡ঢ়گʹͳΓ·͢ɻ
    ิ୊ 10 ʹ͓͚Δ ω ͸ɺࠓͷ৔߹ ω = −1 ͱͳΔͷͰɺF(ω) = F Ͱ͋Δ఺ʹ஫ҙ͍ͯͩ͘͠͞ɻैͬͯɺิ
    ୊ 10 ͷূ໌ͷखଓ͖Λࢥ͍ग़͢ͱɺଟ߲ࣜ
    x2ʵξ2 ∈ F[x] (ξ ∈ E, ξ2 ∈ F) (64)
    ͷࠜ ξ ∈ E = Q(α1
    , α2
    ) ͕ଘࡏͯ͠ɺE = F(ξ) ͱͳΔ͸ͣͰ͢ɻ͜ͷ ξ ͸ɺͦΕࣗ਎͸ F ͷݩͰ͸ͳ͍ͷ
    Ͱɺα1
    ͱ α2
    Λࣗ༝ͳܗͰؚΉ͜ͱ͕Ͱ͖·͕͢ɺ2 ৐ͨ͠ࡍʹ͸ F ͷݩɺͭ·Γɺα1
    ͱ α2
    ͷରশࣜʹͳ
    Δඞཁ͕͋Γ·͢ɻ͜ͷΑ͏ͳ౎߹ͷΑ͍૊Έ߹Θͤʹɺα1
    ͱ α2
    ͷ൓ରশ͕ࣜ͋Γ·͢ɻ࣮ࡍɺ
    ξ = α1
    − α2
    (65)
    ͱఆٛ͢Δͱɺ
    ξ2 = α2
    1
    − 2α1
    α2
    + α2
    2
    = (α1
    + α2
    )2 − 4(α1
    α2
    ) (66)
    ͕੒Γཱͪ·͢ɻ࠷ޙͷදࣜΑΓɺξ2 ∈ F = Q(α1
    + α2
    , α1
    α2
    ) Ͱ͋Δ͜ͱ͕෼͔Γ·͢ɻ͞Βʹ·ͨɺF(ξ)
    ͷൣғͰ͋Ε͹ɺ࣍ͷΑ͏ʹɺα1
    ͱ α2
    Λߏ੒͢Δ͜ͱ͕Ͱ͖·͢ɻ
    α1
    =
    1
    2
    {(α1
    + α2
    ) + ξ}
    α2
    =
    1
    2
    {(α1
    + α2
    ) − ξ}
    ͜ΕΑΓɺ͔֬ʹ E = F(ξ) ͕੒Γཱͭ͜ͱ͕෼͔Γ·͢ɻ
    ͦͯ͠ɺ(66) ͷදࣜΑΓɺξ ͸ɺ
    x2ʵ (α1
    + α2
    )2 − 4(α1
    α2
    )
    51

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  52. ͷࠜͰ͋Δ͜ͱ͔Βɺ
    ξ = (α1
    + α2
    )2 − 4(α1
    α2
    )
    ͱදΘ͢ͱɺΑ͘஌ΒΕͨ 2 ࣍ํఔࣜͷղͷެ͕ࣜಘΒΕ·͢ɻ
    α1
    =
    1
    2
    (α1
    + α2
    ) + (α1
    + α2
    )2 − 4(α1
    α2
    )
    α2
    =
    1
    2
    (α1
    + α2
    ) − (α1
    + α2
    )2 − 4(α1
    α2
    )
    ͳ͓ɺ͜͜ͰಘΒΕ֦ͨେମ E = Q(α1
    , α2
    ) ͸ɺ͋͘·Ͱɺ༗ཧ਺ମ Q ʹ α1
    , α2
    ͱ͍͏ه߸Λܗࣜతʹ
    Ճ͑ͨମͰ͕͢ɺQ ͷ୅਺తดแ͸ෳૉ਺ମ C ʹಉܕʹͳΔͱ͍͏ࣄ࣮͔ΒɺC ͷ෦෼ମͱಉܕʹͳΔ͜ͱ
    ͕อূ͞Ε·͢ɻ۩ମతͳ਺஋܎਺Λ࣋ͭ 2 ࣍ํఔࣜʹରͯ͠ɺ্هͷެࣜΛ౰ͯ͸ΊͯղΛܭࢉ͢Δࡍ͸ɺ
    E Λ C ʹಉܕʹຒΊࠐΜͰɺE → C ͷಉܕࣸ૾Λద༻͍ͯ͠Δ͜ͱʹͳΓ·͢ɻ
    ·ͨɺ্هͷखଓ͖ʹ͓͍ͯɺ(65) Λܾఆ͢Δ෦෼͍ͭͯ͸ɺ΍΍ൃݟతͳख๏Λ༻͍·ͨ͠ɻ͜ͷ෦෼ʹ
    ͍ͭͯ͸ɺద੾ͳ৚݅Λຬͨ͢ ξ Λ௚઀తʹܾఆ͢Δํ๏͕͋ΔΘ͚Ͱ͸͋Γ·ͤΜ͕ɺิ୊ 10 ͷূ໌Ͱ༻
    ͍ͨɺ८ճ܈ͷੜ੒ݩʹର͢Δݻ༗ํఔࣜΛख͕͔Γʹ͢Δ͜ͱ͕Ͱ͖·͢ɻࠓͷ৔߹ɺS2
    ͷੜ੒ݩ σ ͸ɺ
    α1
    ͱ α2
    ͷೖΕସ͑ૢ࡞Ͱ͋Γɺn = 2 ʹର͢Δݪ࢝ n ৐ࠜ͸ ω = −1 ͱͳΓ·͢ɻैͬͯɺ(64) Λ༩͑Δ
    ξ ͸ɺݻ༗ํఔࣜ
    σ(ξ) = −ξ
    ͷղͱͯ͠ಘΒΕ·͢ɻ͜Ε͸ɺξ ͸ α1
    ͱ α2
    ͷ൓ରশࣜͰ͋Δ͜ͱΛ͓ࣔͯ͠Γɺ͜Ε͕ (65) Λൃݟ͢Δ
    ώϯτͱͳΓ·͢ɻ
    8.3 3 ࣍ํఔࣜͷղͷެࣜ
    લઅͷٞ࿦Λ n = 3 ͷ৔߹ʹ֦ுͯ͠Έ·͠ΐ͏ɻα1
    , α2
    , α3
    Λࠜͱ͢Δ 3 ࣍ଟ߲ࣜΛܗࣜʹల։͢Δͱɺ
    ͜ΕΒ 3 ͭͷݩͷجຊରশࣜΛ܎਺ͱ͢Δଟ߲ࣜ
    (x − α1
    )(x − α2
    )(x − α3
    ) = x3 − px2 + qx − r
    ͕ಘΒΕ·͢ɻ͜͜Ͱɺα1
    , α2
    , α3
    ͷجຊରশࣜΛ
    p = α1
    + α2
    + α3
    q = α1
    α2
    + α2
    α3
    + α3
    α1
    (67)
    r = α1
    α2
    α3
    ͱఆٛ͠·ͨ͠ɻैͬͯɺ
    E = Q(α1
    , α2
    , α3
    )
    F = Q(p, q, r)
    ͱͯ͠ɺ
    Aut(E/F) ∼
    = S3
    ͱ͍͏ؔ܎͕ಘΒΕ·͢ɻ͜Ε͕Մղ܈Ͱ͋Ε͹ɺ࢛ଇԋࢉͱ΂͖ࠜʹΑͬͯɺجຊର৅͔ࣜΒ α1
    , α2
    , α3
    Λ
    ٻΊΔखଓ͖͕ଘࡏ͢Δ͜ͱʹͳΓ·͢ɻͦͯ͠ɺ࣮ࡍɺ3 ࣍ͷରশ܈ S3
    ͸ɺ࣍ͷਖ਼ن෦෼܈ͷྻʹ͓͍ͯ
    Մղ܈ͱͳΓ·͢ɻ
    Aut(E/F) ∼
    = S3
    ⊃ A3
    ⊃ {1} ∼
    = Aut(E/E) (68)
    52

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  53. ͜͜ʹɺA3
    ͸ 3 ࣍ͷަ୅܈ʢۮஔ׵ͷΈͷ෦෼܈ʣʹͳΓ·͢ɻΨϩΞ܈ͷ෦෼܈ͷྻ͸ɺ֦େ E/F ͷத
    ؒମͷྻʹରԠ͢Δ΋ͷͰͨ͠ͷͰɺ͜Ε͸ɺ
    E ⊃ M ⊃ F (69)
    ͱ͍͏தؒମ M ͷଘࡏΛ͓ࣔͯ͠Γɺ
    Aut(E/M) ∼
    = A3
    (70)
    ͱ͍͏ରԠ͕੒Γཱͪ·͢ɻͦͯ͠ɺ֦େ M/F ʹ൐͏ΨϩΞ܈ Aut(M/F) ͸ɺ৒༨܈ͷಉܕؔ܎Λ༻͍ͯɺ
    Aut(M/F) ∼
    = Aut(E/F)/Aut(E/M) ∼
    = S3
    /A3

    = S2
    (71)
    ͱܭࢉ͞Ε·͢ɻ(68) ͱ (69) ͸ɺఆཧ 18 ͷূ໌ʹ͓͚Δɺ(60) ͱ (59) ʹରԠ͓ͯ͠Γɺ͜ͷূ໌ͱಉ͡ख
    ॱʹΑΓɺ(69) ͷ֦େྻΛ΂͖֦ࠜେͷྻʹ֦ு͍ͯ͘͜͠ͱ͕Ͱ͖·͢ɻࠓͷ৔߹ɺྡΓ߹͏܈ͷΨϩΞ
    ܈ (70) ͱ (71) ͸͢Ͱʹ८ճ܈ʹͳ͍ͬͯΔͷͰɺ͜ΕΒͷ֦େ͕ͦͷ··΂͖֦ࠜେͱͳΓ·͢ɻͭ·Γɺ
    ֦େ M/F ͱ֦େ E/M ͷͦΕͧΕʹରԠ͢Δ΂͖֦ࠜେΛ۩ମతʹߏ੒͢Δ͜ͱͰɺ3 ࣍ํఔࣜͷղͷެ
    ࣜɺ͢ͳΘͪɺα1
    , α2
    , α3
    ͷରশࣜͰ͋Δ p, q, r ͔Βɺα1
    , α2
    , α3
    Λݸผʹߏ੒͢Δखଓ͖͕ಘΒΕ·͢ɻ
    ͦΕͰ͸ɺରԠ͢Δ΂͖֦ࠜେΛ࣮ࡍʹ֬ೝ͍͖ͯ͠·͢ɻ·ͣɺ֦େ M/F ʹ͍ͭͯ͸ɺ
    [M : F] = |S2
    | = 2
    ͱͳΔͷͰɺ2 ࣍ํఔࣜͷ࣌ͱಉٞ͡࿦Ͱɺ
    x2 − δ2 ∈ F[x] (δ ∈ E, δ2 ∈ F)
    ͷࠜ δ ∈ E Λ༻͍ͯɺ
    M = F(δ)
    ͕੒ཱ͠·͢ɻ͜͜Ͱɺ۩ମతͳ δ ͷදࣜΛݟ͚ͭΔͨΊʹɺલઅͷ࠷ޙʹઆ໌ͨ͠ݻ༗ํఔࣜΛར༻͠·
    ͢ɻࠓɺ
    Aut(M/F) ∼
    = S3
    /A3

    = S2
    = {1, σ}
    ͸ɺS3
    ʹؚ·ΕΔஔ׵Λحஔ׵ σ ͱۮஔ׵ 1 ͷ 2 छྨʹ෼ྨͨ͠৒༨܈Ͱ͋Γɺω = −1 Ͱ͋Δ͜ͱ͔Βɺ
    δ ʹର͢Δݻ༗ํఔࣜ͸ɺ
    σ(δ) = −δ
    ͱͳΓ·͢ɻͭ·Γɺδ ͸ɺ{α1
    , α2
    , α3
    } ͷ೚ҙͷحஔ׵ʹରͯ͠൓ରশͰ͋Δ͜ͱ͕ཁ੥͞Ε·͢ɻͦ͜Ͱɺ
    δ = (α1
    − α2
    )(α2
    − α3
    )(α3
    − α1
    ) (72)
    ͱ͍͏ͦΕͧΕͷݩʹ͍ͭͯ൓ରশͳ૊Έ߹ΘͤΛߟ͑Δͱɺ͜ͷ৚݅Λຬͨ͠·͢ɻ͜ͷ࣌ɺδ2 ͸ۮஔ׵Ͱ
    ΋حஔ׵Ͱ΋ෆมͰ͋Γɺݴ͍׵͑Δͱ Aut(E/F) ∼
    = S3
    ͰෆมʹͳΓ·͢ɻ͜ΕΑΓɺ͔֬ʹ δ2 ∈ F ͱͳ
    Δ͜ͱ͕෼͔Γ·͢ɻ
    ʢE ͸ F ্ͷ࠷খ෼ղମͰ͋Δ͜ͱ͔Β E/F ͸ΨϩΞ֦େͰ͋ΓɺEAut(E/F ) = F
    ͕੒Γཱͪ·͢ɻ
    ʣ͋Δ͍͸ɺδ2 ͸ α1
    , α2
    , α3
    ʹ͍ͭͯͷରশࣜʹͳΔͷͰɺجຊର৅ࣜͰදݱͰ͖Δ͜ͱ͸
    ࣗ໌Ͱɺ͜Ε͔Β΋ δ2 ∈ F ͱͳΔ͜ͱ͕ݴ͑·͢ɻҎ্ʹΑΓɺ(72) Ͱఆٛ͞ΕΔ δ Λ༻͍ͯɺ΂͖֦ࠜେ
    M = F(δ) Λߏ੒͢Δ͜ͱ͕Ͱ͖·ͨ͠ɻ
    ࣮ࡍʹɺδ2 Λ α1
    , α2
    , α3
    ͷجຊର৅ࣜͰॻ͖ද͢ͷ͸ɺͦΕ΄Ͳ؆୯ͳ࡞ۀͰ͸͋Γ·ͤΜ͕ɺܭࢉաఔ
    ͸লུͯ͠ɺͱʹ͔͘ɺ࠷ऴ݁ՌΛࣔ͢ͱ࣍ͷΑ͏ʹͳΓ·͢ɻ
    δ2 = −4p3r − 27r2 + 18pqr − 4q3 + p2q2 (73)
    53

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  54. (72) ͔Βܭࢉ͞ΕΔ δ2 ͱɺ্ࣜͷӈลʹ (67) Λ୅ೖͯ͠ల։ͨ݁͠Ռ͕Ұக͢Δ͜ͱ͸ɺ௚઀ܭࢉͰ֬ೝ
    Ͱ͖ΔͰ͠ΐ͏ɻ
    ࣍͸ɺ֦େ E/M ʹ͍ͭͯߟ͑·͢ɻ·ͣɺ(70) ΑΓɺ
    [E : M] = |A3
    | = 3
    ͱͳΔͷͰɺ͜Ε͸ɺิ୊ 10 Ͱ n = 3 ͱஔ͍ͨ৔߹ͱಉ͡ঢ়گʹͳΓ·͢ɻ΋͏গ͠ਖ਼֬ʹݴ͏ͱɺఆཧ 18
    ͷূ໌ͰߦͬͨΑ͏ʹɺ1 ͷݪ࢝ 3 ৐ࠜΛ ω ͱͯ͠ɺ֦େ E(ω)/M(ω) ʹରͯ͠ิ୊ 10 Λద༻͠·͢ɻͦͷ
    ݁ՌɺM(ω) ্ͷଟ߲ࣜ
    x3 − u3 ∈ M(ω)[x] (u ∈ E(ω), u3 ∈ M(ω)) (74)
    ͷࠜ u Λ༻͍ͯɺM(ω) ͔Β E(ω) ΁ͷ΂͖֦ࠜେ
    E(ω) = M(u, ω)
    ͕੒ཱ͠·͢ɻ͜ΕʹΑΓɺશମͱͯ͠ɺ
    E(ω) = M(u, ω) ⊃ M(ω) ⊃ M = F(δ) ⊃ F
    ͱ͍͏΂͖֦ࠜେͷྻ͕׬੒͠·͢ɻͦΕͰ͸ɺิ୊ 10 ͷূ໌ʹैͬͯɺ(74) ͷ৚݅Λຬͨ͢ u ͷ۩ମతͳ
    දࣜΛݟ͚ͭ·͠ΐ͏ɻ͜͜Ͱ΋·ͨɺݻ༗ํఔࣜΛར༻͢Δ͜ͱʹͳΓ·͢ɻ·ͣɺఆཧ 18 ͷূ໌ͷதͰ
    ࣔͨ͠Α͏ʹɺ
    Aut(E(ω)/M(ω)) ∼
    = Aut(E/M) ∼
    = A3
    Ͱ͋ΓɺA3
    ͷੜ੒ݩͱͯ͠ɺ
    σ = (1, 2, 3) ∈ A3
    Λબ୒͢Δ͜ͱ͕Ͱ͖·͢*12ɻैͬͯɺσ ʹର͢Δݻ༗ํఔࣜ
    σ(u) = ωu (75)
    Λຬͨ͢ u ͕ൃݟͰ͖Ε͹Α͍͜ͱʹͳΓ·͢ɻ΍΍ఱԼΓతͰ͕͢ɺ͜͜Ͱ͸ɺu Λ α1
    , α2
    , α3
    ͷ 1 ࣍݁
    ߹Ͱ͋ΔͱԾఆͯ͠ɺ
    u = a1
    α1
    + a2
    α2
    + a3
    α3
    (a1
    , a2
    , a3
    ∈ Q)
    ͱஔ͍ͯΈ·͢ɻ͜ΕΛ (75) ʹ୅ೖͯ͠ܭࢉ͢Δͱɺ
    a2
    = ω2a1
    a3
    = ωa1
    ͱ͍͏ؔ܎͕ಘΒΕΔͷͰɺu ͸࣍ͷΑ͏ʹܾ·Γ·͢ɻ
    u = α1
    + ω2α2
    + ωα3
    (76)
    ݻ༗ϕΫτϧʹ͸ఆ਺ഒͷࣗ༝౓͕͋ΔͷͰɺ͜͜Ͱ͸ɺa1
    = 1 ͱ͍ͯ͠·͢ɻͦͯ͠ɺ͜ͷ࣌ɺ
    σ(u3) = {σ(u)}3 = u3
    *12 (1, 2, 3) ͸ɺα1 → α2, α2 → α3, α3 → α1
    ͱ͍͏८ճஔ׵Λද͠·͢ɻ
    54

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  55. Ͱ͋Δ͜ͱ͔Βɺu3 ͸ Aut(E(ω)/M(ω)) Ͱݻఆ͞Ε͓ͯΓɺ͔֬ʹ
    u3 ∈ M(ω)
    ͱͳ͍ͬͯ·͢ɻ
    ʢE(ω) ͸ (74) ͷ M(ω) ্ͷ࠷খ෼ղମͰ͋Δ͜ͱ͔Β E(ω)/M(ω) ͸ΨϩΞ֦େͰ͋Γɺ
    E(ω)Aut(E(ω)/M(ω)) = M(ω) ͕੒Γཱͪ·͢ɻ
    ʣͦͯ͠ɺu3 ͸ɺ
    M(ω) = F(ω, δ) = Q(p, q, r, ω, δ)
    ͷݩͰ͢ͷͰɺp, q, r, ω, δ ͷ૊Έ߹ΘͤͰදݱͰ͖Δ͸ͣͰ͢ɻͪ͜Β΋ಋग़ํ๏͸ͦΕ΄Ͳ؆୯Ͱ͸͋Γ·
    ͤΜ͕ɺͱʹ͔͘ɺ࠷ऴ݁ՌΛࣔ͢ͱ࣍ͷΑ͏ʹͳΓ·͢ɻ͜͜Ͱ͸ɺω2 + ω + 1 = 0 ͷؔ܎Λ༻͍ͯɺω2
    ͷ߲Λফڈͯ͋͠Γ·͢ɻ
    u3 = p3 −
    9
    2
    (pq − 3r) + 3
    1
    2
    + ω δ
    Ҏ্Ͱ΂͖֦ࠜେͷྻ͸ߏ੒Ͱ͖·͕ͨ͠ɺҰൠతͳղͷެࣜΛಘΔʹ͸ɺ΋͏গ͠࡞ۀ͕ඞཁͰ͢ɻ2 ࣍
    ํఔࣜͷ৔߹ɺξ ͱ α1
    + α2
    ͷ૊Έ߹ΘͤͰɺα1
    ͱ α2
    Λ໌ࣔతʹߏ੒͢Δ͜ͱ͕Ͱ͖·͕ͨ͠ɺࠓͷ৔߹ɺ
    (76) ͷ u ͱ α1
    + α2
    + α3
    Λ૊Έ߹ΘͤΔ͚ͩͰ͸ɺα1
    , α2
    , α3
    Λߏ੒͢Δ͜ͱ͸Ͱ͖·ͤΜɻ΋͏ 1 ͭɺu
    ͱಉ༷ͷ໾ׂΛՌͨ͢ݩ͕ඞཁͰ͢ɻͦ͜Ͱɺu Λൃݟͨ͠ํ๏Λࢥ͍ग़͢ͱɺ͜Ε͸ɺݻ༗ํఔࣜ (75) ͷ
    ղͱͯ͠ಘΒΕ·ͨ͠ɻ͜ͷ࣌ɺݻ༗஋ͱͯ͠ ω Λબͼ·͕ͨ͠ɺิ୊ 10 ͷূ໌ͷྲྀΕΛߟ͑Δͱɺଞͷݻ
    ༗஋Λ༻͍ͯ΋ಉ༷ͷٞ࿦͕Ͱ͖Δ͜ͱ͕෼͔Γ·͢ɻ
    ࠓͷ৔߹ɺσ ͷݻ༗஋ʹ͸ɺ{1, ω, ω2} ͷ 3 छྨ͕͋ΔͷͰɺͦΕͧΕʹର͢Δݻ༗ํఔࣜΛղ͍ͯɺରԠ
    ͢Δݻ༗ϕΫτϧΛٻΊͯΈ·͠ΐ͏ɻ(76) ΛٻΊͨ࣌ͱಉ͡ํ๏Ͱܭࢉ͢Δͱɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ
    a = α1
    + α2
    + α3
    u = α1
    + ω2α2
    + ωα3
    (77)
    ν = α1
    + ωα2
    + ω2α3
    u ͱಉٞ͡࿦ʹΑΓɺa3, u3, ν3 ͸ɺ͢΂ͯ M(ω) = F(ω, δ) = Q(p, q, r, ω, δ) ͷݩͱͳΓɺp, q, r, ω, δ ͷ૊
    Έ߹ΘͤͰදݱ͢Δ͜ͱ͕Ͱ͖·͢ɻν ͸ɺu ʹ͓͍ͯ ω ͱ ω2 ͷ໾ׂΛஔ͖׵͑ͨ΋ͷͰ͋Δ఺ʹ஫ҙ͢
    Δͱɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ
    a3 = p3
    u3 = p3 −
    9
    2
    (pq − 3r) + 3
    1
    2
    + ω δ (78)
    ν3 = p3 −
    9
    2
    (pq − 3r) + 3
    1
    2
    + ω2 δ
    ͜ΕͰղͷެࣜΛಋ͘४උ͕Ͱ͖·ͨ͠ɻ·ͣɺ(77) Λ α1
    , α2
    , α3
    ʹ͍ͭͯٯղ͖͢Δͱɺ࣍ͷؔ܎͕ಘ
    ΒΕ·͢ɻ
    α1
    =
    1
    3
    (a + u + ν)
    α2
    =
    1
    3
    (a + ωu + ω2ν)
    α3
    =
    1
    3
    (a + ω2u + ων)
    55

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  56. ͜Εʹɺ(78) ͔ΒಘΒΕΔؔ܎
    a = p
    u = 3 p3 −
    9
    2
    (pq − 3r) + 3
    1
    2
    + ω δ
    ν = 3 p3 −
    9
    2
    (pq − 3r) + 3
    1
    2
    + ω2 δ
    Λ୅ೖ͠ɺ͞Βʹɺ(73) ͔ΒಘΒΕΔؔ܎
    δ = −4p3r − 27r2 + 18pqr − 4q3 + p2q2
    Λ୅ೖ͢Ε͹ɺα1
    , α2
    , α3
    Λ p, q, r Ͱ໌ࣔతʹॻ͖ද͢͜ͱ͕Ͱ͖·͢ɻ͜Ε͕ɺ3 ࣍ํఔࣜͷղͷެࣜͱ͍
    ͏͜ͱʹͳΓ·͢ɻ
    ࢀߟจݙ
    [1] ࠷௿ݶͷ Galois ཧ࿦ (ver.2014.01.07) http://staff.miyakyo-u.ac.jp/ k-taka2/pdf/galois.pdf
    [2] ΨϩΞཧ࿦ೖ໳ϊʔτʢৄࡉʣ http://www.tsuyama-ct.ac.jp/matsuda/galois/gals.pdf
    [3] ෺ཧͷ͔͗ͬ͠Άʢ୅਺ֶʣ http://hooktail.org/misc/index.php?%C2%E5%BF%F4%B3%D8
    [4] ؀ͷ४ಉܕఆཧ http://rikei-index.blue.coocan.jp/daisu/zyundoukeikan.html
    56

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