## Splitting Fields

K/F, L/F extension fields, Ξ± β K algebraic over F, Ο :K β L F-homomorphism of fields β Ο(Ξ±) is algebraic over F. f(x) β F[x], K/F s.f. if f(x) = c(x -Ξ±β)(x -Ξ±β) Β·Β·Β· (x -Ξ±β), Ξ±α΅’ β K, K = F(Ξ±β, Β·Β·Β·, Ξ±β). Existence of Splitting Fields, F field, f(x) β F[x] β β s.f. E for f(x) over F. Th, F field, p(x) β F[x] irreducible polynomial over F, deg(p(x)) = n. If a is a root of p(x) in some extension E of F, then F(a) β F[x]/β¨p(x)β© & F(a) is a vector space over F with a basis {1, a, Β·Β·Β·, aβΏβ»ΒΉ}. Degree of Splitting Field. F field, p β F[x], deg(p) = n β βK/F s.f., [K:F] β€ n!

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3151 words,π15 minutes read, May 14, 2022.