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 40