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An algebraic extension in which every element is separable over the base field.
Let be an algebraic field extension.
Definition (separable extension). The extension is separable if every element is a separable element over .
When is finite, separability admits useful equivalent formulations. For example, if is an algebraic closure of , then is separable iff there are exactly distinct -embeddings . Separability behaves well in towers, as summarized in separability in towers.
Let and be fields. A field extension of is a pair where is an injective field homomorphism. In practice one identifies with its image , writes simply , and denotes the extension by .
Let be a field extension and let be an algebraic element over . Let be the minimal polynomial of over .
Definition (separable element). The element is separable over if the polynomial has no repeated roots in a splitting field (equivalently, in an algebraic closure of ). In other words, has distinct roots.
A convenient algebraic criterion is:
where is the formal derivative. The link between separability and distinct roots is recorded in separable polynomials have distinct roots.
Let be a field.
Definition (algebraic closure). An algebraic closure of is a field equipped with an inclusion such that:
In particular, every splits in , so contains a splitting field for every polynomial over .
Let and be fields. A field embedding is a ring homomorphism such that and is injective.
More generally, given a field extension and a field containing , a -embedding of into is a field embedding whose restriction to is the identity map. In this setting, is also called a -homomorphism. If is bijective, it is a field automorphism of (and if it fixes , a -automorphism).
Consider a tower of fields with algebraic. Separability behaves transitively:
Theorem (tower property for separability).
Let be an algebraic field extension.
Definition (normal extension). The extension is normal if for every irreducible polynomial , the condition “ has a root in ” implies “ splits into linear factors in ”.
Let be an algebraic field extension.
Definition (Galois extension). The extension is Galois if it is both