perfect field



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.


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

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

Revised on July 6, 2015 10:10:23 by Todd Trimble (