Separability—Section 10.5
Sarah Vilardi
April 12, 2011
Abstract Algebra II
From Thursday…
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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.
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(f+g)’(x)=f’(x)+g’(x)

(fg)’(x)=f(x)g’(x)+f’(x)g(x)
Lemma 10.16
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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.
Definition

A field F is said to have characteristic 0 if n1F ≠ 0F for every positive n.
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.
Theorem 10.18
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If K is a finitely generated separable extension field of F, then K = F(u) for
some u in K.
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This proof is a beast…an outline will help us!
Theorem 10.18 Outline
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By hypothesis, K = F(u1,…,un). Proof is by induction on n.
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Work with n = 2 case, where K = F(v, w).
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Establish preliminary assumptions (min. polys, roots, splitting fields, etc.).
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Claim: K = F(u). Prove that w is an element of F(u).
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Let r(x) be the minimal polynomial of w over F(u).
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Show that r(x) is linear.
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Prove that K = F(v, w) = F(u).