nLab
perfect field

Contents

Definition

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

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

A field, FF, of characteristic pp is perfect if every element of FF is a ppth 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

Revised on July 21, 2017 12:52:25 by Anonymous (86.147.121.29)