Find a field $K$ with $\operatorname{char}K=2$ which has an algebraic extension $F$ containing a particular subset

Question

I am looking for an example of a field $K$ of characteristic 2, which, for some algebraic extension $F$ of $K$, $F$ contains a subset $S$ such that:

a) $K\subseteq S$,

b) $S$ is a vector space over $K$,

c) $s^n\in S$ for all $s\in S$ and all positive integers $n$,

d) $S$ is $\textit{not}$ a subfield of $F$.

This is from Beachy and Blair Abstract Algebra, which asks to find an example of a field $K$ which is a counter example to exercise 6.2.11 (no such $S$ exists if $\operatorname{char}K\ne2$).

Thank you for any help and I apologize if my grammar is bad, I am learning English. Thank you.

Answer 1

Take a field $K$ of characteristic 2, let $L=K(x,y), F=K(u,v)$ rational functions in two variables and consider the inclusion $L\subset F$ given by $x\mapsto u^2, y\mapsto v^2$. Let $S$ be the $L$ vector space generated by $1,u,v$. Then $S$ is not a subfield, since $uv\not\in S$. But, easy to check that $s^n\in S$ for any positive integer $n$ and $s\in S$.