ガロア理論入門

Transcript

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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<···<ik αi1 · · · αik (62) ۩ମతʹ͸ɺ࣍ͷΑ͏ͳܗʹͳΓ·͢ɻ s1 = α1 + · · · + αn s2 = α1 α2 + · · · + αn−1 αn (63) . . . sn = α1 · · · αn 49
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
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
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
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
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
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
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