If $F$ is an extension field of the field $K$ such that $[F:K] =1$, then $F=K$

Question

Suppose $F$ is an extension field of the field $K$ such that $[F:K] =1$. How to prove that $F=K$?

Thank you for your time and help.

Answer 1

Just note that one would tend not to say $F=K$ but rather $F \cong K$.

To think of why this should be so, as a hint, think of a one dimensional vector space over $\mathbb{R}$, say $V$. Then $V$ "looks like" $\mathbb{R}v$. Can you think of an obvious isomorphism of this to $\mathbb{R}$?

Answer 2

Two simple proofs:

• $F$ and $K$ are both $F$-subspaces of $K$. Since $F\subseteq K$ and they have the same dimension, they must be equal.

• If $K$ has dimension $1$ as an $F$-subspace, then $K=Fk$ for every $k\ne0$. In particular, with $k=1$, we get $K=F$.