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 , #m/def/field i.e.

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

Properties

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.
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

#state/tidy | #lang/en | #SemBr