## Splitting Fields

K/F, L/F extension fields, α ∈ K algebraic over F, σ :K → L F-homomorphism of fields ⇒ σ(α) is algebraic over F. f(x) ∈ F[x], K/F s.f. if f(x) = c(x -α₁)(x -α₂) ··· (x -αₙ), αᵢ ∈ K, K = F(α₁, ···, αₙ). Existence of Splitting Fields, F field, f(x) ∈ F[x] ⇒ ∃ s.f. E for f(x) over F. Th, F field, p(x) ∈ 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) ≋ F[x]/⟨p(x)⟩ & F(a) is a vector space over F with a basis {1, a, ···, aⁿ⁻¹}. Degree of Splitting Field. F field, p ∈ F[x], deg(p) = n ⇒ ∃K/F s.f., [K:F] ≤ n!