I have another abstract algebra question. I stated it in the title, but here it is in more detail:
Let $F\subseteq R \subseteq E$ where $E$ is an algebraic extension of the field $F$. If $R$ is an $F$-subspace of $E$, $\text{char}R \ne 2$, and $u\in R$ implies that $u^k\in R$ for each $k\geq 2$, show that $F$ is a field.
I've already gotten that, if $R$ is a subring of $E$, then it is a field. So, all I need to do is show that $R$ is closed under multiplication. I think it has to do with minimal polynomials, especially because of the $u^k$ assumption, but I can't figure out how to get $r_1 r_2\in R$ if $r_1,r_2\in R$. Thanks in advance!