View source for Perfect field

A [[Field|field]] $k$ over which every polynomial is separable. In other words, every algebraic extension of $k$ is a [[Separable extension|separable extension]]. All other fields are called ''imperfect''. Every field of characteristic 0 is perfect. A field $k$ of finite characteristic $p$ is perfect if and only if $k = k^p$, that is, if raising to the power $p$ is an automorphism of $k$. [[Finite field]]s and [[algebraically closed field]]s are perfect. An example of an imperfect field is the field $\mathbb{F}_q(X)$ of rational functions over the field $\mathbb{F}_q$, where $\mathbb{F}_q$ is the field of $q = p^n$ elements. A perfect field $k$ coincides with the field of invariants of the group of all $k$-automorphisms of the algebraic closure $\bar k$ of $k$. Every algebraic extension of a perfect field is perfect.

For any field $k$ of characteristic $p>0$ with algebraic closure $\bar k$, the field
$$
k^{p^{-\infty}} = \bigcup_n k^{p^{-n}} \subset \bar k
$$
is the smallest perfect field containing $k$. It is called the ''perfect closure'' of the field $k$ in $\bar k$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Algèbre" , Masson  (1981)  pp. Chapts. 4–5</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR></table>