Are two subfields of the same finite field equal if they have the same size?

Question

Suppose $K$ is a finite field and $F_1,F_2$ are subfields of $K$ such that $|F_1|=|F_2|.$ Does this imply that $F_1=F_2$?

Answer 1

Let $n=|F_1|=|F_2|$.

Consider the multiplicative groups $F_1^*$ and $F_2^*$.

We have $|F_1^*|=n-1$, so the $n-1$ nonzero elements of $F_1$ satisfy $x^{n-1}=1$, hence must be all the roots in $K$ of $x^{n-1}-1$.

But also $|F_2^*|=n-1$, so the $n-1$ nonzero elements of $F_2$ also satisfy $x^{n-1}=1$, hence the set of nonzero elements of $F_2$ is equal to the set of nonzero elements of $F_1$.

It follows that $F_1=F_2$.

Answer 2

A finite field of characteristic $p$ and with $p^n$ elements is a splitting field of the polynomial $\;X^{p^n}-X$. There can be only one splitting field within a given field, because a polynomial with degree $p^n$ over a field cannot have more than $p^n$ roots.