Regarding some field notation

Question

Let $K$ be a finite field of charastic $p$. I've seen the notation $K^{p}$ used to reference a subfield of $K$ (Specifically in Lemma 1 of Serre's A Course in Arithmetic). Would this subfield be the set of all elements of $K$ raised to the $p$th power?

Answer 1

According to the formulation in the book ...

Lemma. If $\operatorname{char}(K)=p$, the map $\sigma\colon x\mapsto x^p$ is an isomorphism onto one of its subfields $K^p$.

... the lemma is also used as a definition for the notation $K^p$. That is, the lemma claims (and the subsequent proof shows) that $K$ has an interesting subfield, and it introduces the notation $K^p$ for this subfield. And yes, the lemma makes $K^p$ the subfield consisting of all $p$th powers.