TITLE: A finite field extension of $\mathbb R$ is either $\mathbb R$ or isomorphic to $\mathbb C$
QUESTION [14 upvotes]: Let $F$ be a field containing $\mathbb R$ with the property that $\dim_{\mathbb R}F < \infty.$ Then either $F \cong \mathbb R$ or $F \cong \mathbb C.$

I am trying to prove the above statement. I am not supposed to use Frobenius' theorem as it would just spoil the spirit of the current problem. I am told to  show that every $x \in F \setminus \mathbb R$ is a root of some non-zero polynomial in $F[x]$ with leading coefficient $1$ and some other things. But I am not really been able to prove anything so far. Please help. Thanks in advance.

REPLY [12 votes]: By the uniqueness of the algebraic closure, we have an embedding $F \hookrightarrow \mathbb C$, hence we have $\mathbb R \subset F \subset \mathbb C$. The result follows from $[\mathbb C:\mathbb R]=2$, because this excludes the existence of proper intermediate fields.