Separability —Section 10.5

1. Separability—Section 10.5 Sarah Vilardi April 12, 2011 Abstract Algebra II

2. From Thursday… • Let F be a field. A polynomial f(x) in F[x] of degree n is said to be separable if f(x) has n distinct roots in every splitting field. If K is an extension field of F, then an element u in K is separable over F if u is algebraic over F and its minimal polynomial p(x) in F[x] is separable. The extension field K is a separable extension if every element of K is separable over F. • (f+g)’(x)=f’(x)+g’(x) • (fg)’(x)=f(x)g’(x)+f’(x)g(x)

3. Lemma 10.16 • Let F be a field and f(x) be in F[x]. If f(x) and f’(x) are relatively prime in F[x], then f(x) is separable.

4. Definition • A field F is said to have characteristic 0 if n1F ≠ 0F for every positive n.

5. Theorem 10.17 • Let F be a field of characteristic 0. then every irreducible polynomial in F[x] is separable, and every algebraic extension field K of F is a separable extension.

6. Theorem 10.18 • If K is a finitely generated separable extension field of F, then K = F(u) for some u in K. • This proof is a beast…an outline will help us!

7. Theorem 10.18 Outline • By hypothesis, K = F(u1,…,un). Proof is by induction on n. • Work with n = 2 case, where K = F(v, w). • Establish preliminary assumptions (min. polys, roots, splitting fields, etc.). • Claim: K = F(u). Prove that w is an element of F(u). • Let r(x) be the minimal polynomial of w over F(u). • Show that r(x) is linear. • Prove that K = F(v, w) = F(u).