perfect field

A perfect field is one where the irritating pathologies of positive characteristic simply do not arise. Over such a field, every algebraic extension behaves as nicely as extensions of the rationals: no minimal polynomial ever has a repeated root, no element is 'inseparable', and Galois theory works without extra hypotheses. Most fields you ever meet — the rationals, the reals, the complexes, all finite fields — are perfect, which is why these subtleties stay hidden until you deliberately go looking.

Precisely, a field K is perfect if every algebraic extension of K is separable, equivalently if every irreducible polynomial over K is separable. There is a crisp criterion: a field of characteristic 0 is always perfect, and a field K of characteristic p is perfect if and only if the Frobenius endomorphism x to x^p is surjective, i.e. every element of K has a p-th root in K.

The standard imperfect field is the rational function field F_p(t): the element t has no p-th root inside it, so Frobenius is not surjective, and x^p - t is irreducible but inseparable. Passing to the perfect closure (adjoining all higher p-power roots) repairs this. Perfection is exactly the hypothesis that lets you forget about separability entirely.