Field theory MOC

Separable degree of an extension

Let be an algebraic extension. The separable degree of is given by the number of embeddings of this extension into the algebraic closure , field i.e.

and is nonzero assuming Zorn’s lemma, see Embedding an algebraic extension into an algebraically closed field.

Properties

1. Let be a simple algebraic extension. Then equals the number of distinct roots in of the minimal polynomial , and thus , with equality iff is separable.
2. If is a tower of algebraic extensions then .

Proof of 1–2.

The proof of ^P1 is very similar to that of the Bound on the automorphism group of a finite simple extension. Associate with each the image , which must be a root of . Since completely determines , this correspondence is injective. For surjectivity, let be a root of . Then by ^P1, there exists an isomorphism sending , and composing this with the embedding gives the corresponding , proving ^P1.

For ^P2, suppose is such a tower of extensions, and identify . It is not difficult to see that we have a bijection

by first extending to and then extending to .

Results

• Separability of a finite extension

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