# Contents

## Definition

A field (in the sense of commutative algebra) $F$ is perfect if every algebraic extension of $F$ is separable. In that case, every splitting field extension of $F$ is a Galois extension.

An extension $E/F$ is separable iff every element $\alpha \in E$ is separable, meaning that its irreducible polynomial $f \in F[x]$ (a monic generator of the kernel of $F[x] \to E: x \mapsto \alpha$) has no multiple roots. Of course $f$ has a multiple root only if its derivative satisfies $f'(\alpha) = 0$, which means $f' \in (f)$: by degree considerations this can happen only if $f'$ is the zero polynomial. Notice this cannot happen in characteristic zero.

A field, $F$, of characteristic $p$ is perfect if every element of $F$ is a $p$th power. This property is used in the generalization to perfect rings?.

## Examples

All fields of characteristic zero are perfect, as are all finite fields, all algebraically closed fields, and all algebraic extensions of perfect fields.

An example of a field that isn’t perfect is the field of rational functions over a finite field.

## Related entry

Last revised on July 21, 2017 at 12:52:25. See the history of this page for a list of all contributions to it.