Field Theory

Let a field $K$ have characteristic $p>0$. Then given $a\in K$ we have that $x-a$ is either irreducible or of the form $(x-b{)}^p$ with $b\in K$. Hence $K$ is perfect iff $K=K^p$ where $K^p=\{x\in K:\mathrm{\exists }y\in K.x=y^p\}$.

Let $K<L$ be an algebraic extension. Suppose we have a $K$-monomorphism $\psi :L\to {\overline{K}}^a$. Then we can extend this to a ${\psi }^{\prime }:{\overline{L}}^a\to {\overline{K}}^a$. This follows from an obvious construction using Zorn's lemma and a basic property of extensions to algebraic extensions.