Extensions of degree $1$. My doubt is very simple:
Let $F|K$ be a field extension, if $[F:K]=1$, what can we say about $F$ and $K$? can I say $F=K$?
I'm trying to prove the equality without success.
Thanks in advance

The fact that $[F:K] = 1$ means that as a $K$-vector space, $F$ has a basis of cardinality $1$. Let us suppose this basis is $B = \{k_0\}$.
Since a basis is a spanning set, this means that for ANY $\alpha \in F$, we have: $\alpha = c\cdot k_0$ for some $c \in K$. But the $K$-action (scalar multiplication) of $F$ as an extension field of $K$ is just the $F$-multiplication.
So $\alpha = ck_0$. Now as a subring of $F$, $K$ is closed under multiplication, so we conclude $F \subseteq K$.
Since $F$ is an extension of $K$, we have a priori, $K \subseteq F$, thus the two are equal.
