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 $K^p = \{ x \in K: \exists y \in K. x = y^p\}$.

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