nLab perfect field

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

• perfect ring
• perfectoid field