What is a perfect field, why is perfection important, and what is an example of a nonperfect field?
Solution
A field \(F\) is perfect if every algebraic extension of \(F\) is separable. Equivalently, every irreducible polynomial in \(F[x]\) is separable.
Every field of characteristic \(0\) is perfect, because the derivative of a nonconstant irreducible polynomial cannot vanish identically.
If \(\operatorname{char}F=p>0\), then \[ F\text{ is perfect}\iff F^p=F, \] that is, iff the Frobenius map \[ F\to F,\qquad x\mapsto x^p \] is surjective. Indeed, inseparability in characteristic \(p\) occurs exactly when an irreducible polynomial has zero derivative, hence is a polynomial in \(x^p\).
A standard nonperfect field is \[ F=\FF_p(t). \] The element \(t\) is not a \(p\)th power in \(F\), so Frobenius is not surjective. Equivalently, \[ x^p-t\in F[x] \] is irreducible and inseparable.
Perfection matters because over a perfect field all finite algebraic extensions are separable, so normal finite extensions are automatically Galois and the usual finite Galois correspondence applies without inseparability pathologies.