Slide 40
Slide 40 text
ఆཧ 16 2 छྨͷ֦େ E/Fɺ͓ΑͼɺE′/F ͕͋ΓɺͲͪΒ F ্ͷ n ࣍ଟ߲ࣜ f(x) ͷ࠷খղମʹͳͬ
͓ͯ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
)
͕Γཱͭͷͱ͢Δɻ͜ͷ࣌ɺE ͔Β E′ ͷಉܕࣸ૾Ͱɺ{α1
, · · · , αn
} Λ {β1
, · · · , βn
} ʹஔ͢Δͷ
͕ߏͰ͖Δɻ
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
} ͷॱ൪Λฒସ͑ͨͷͰɺ͜ͷબΛ͏·͘ߦ͏ͱɺ্هͷࣸ૾
͕ͯ͢ମͱͯ͠ͷಉܕࣸ૾ʹͳΔ͜ͱΛ k (0 ≤ k ≤ n) ʹ͍ͭͯͷֶతؼೲ๏Ͱূ໌͢Δɻk = 0 ͷ࣌
ࣗ໌ͳͷͰɺk − 1 ·Ͱཱ͍ͯ͠Δͱͯ͠ɺk (k ≥ 1) ͷ߹Λߟ͑Δɻ
·ͣɺଟ߲ࣜ g(x) ∈ Fk−1
[x] ʹରͯ͠ɺͯ͢ͷʹ τk−1
Λ࡞༻ͨ͠ͷΛ gτk−1 (x) ∈ F′
k−1
[x] ͱද
ه͢Δͱɺτk−1
͕ಉܕࣸ૾ͱ͍͏Ծఆ͔Βɺ࣍ɺଟ߲ࣜͷؒͷʢͱͯ͠ͷʣಉܕࣸ૾ͱͳΔɻ
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
ʹҙ͢Δʣ
ɺf(x) Λ Fk−1
্ͷଟ߲ࣜͱׂͯͬͨ࣌͠ͷΛ q(x) ͱͯ͠ɺ
f(x) = p(x)q(x) (p(x), q(x) ∈ Fk−1
[x])
ͱදΘ͢ɻ͜͜Ͱɺp(αk
) = f(αk
) = 0 ΑΓɺ༨߲߃తʹ 0 ʹͳΔ͜ͱΛ༻͍ͨɻ͜ͷ྆ลʹ τk−1
Λ
࡞༻ͤ͞ΔͱɺF′
k−1
্ͷଟ߲ࣜͷؔͱͯ͠ɺ
f(x) = pτk−1 (x)qτk−1 (x) (pτk−1 (x), qτk−1 (x) ∈ F′
k−1
[x])
͕ಘΒΕΔɻτk−1
F ͷݩಈ͔͞ͳ͍ͷͰɺfτk−1 (x) = f(x) ͱͳΔࣄΛ༻͍ͨɻ͜͜Ͱɺpτk−1 (x) 2
࣍Ҏ্ͷଟ߲ࣜͳͷͰɺE′ ্Ͱ f(x) ΛҼղͨ͠ࡍͷগͳ͘ͱ 2 ͭͷҼ͕ pτk−1 (x) ʹؚ·Ε͓ͯΓɺ
pτk−1 (β′
k
) = 0
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