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

Suppose $\exists a \in K \setminus F$. Then $1, a$ are $F$-linearly independent. Proof: If $f_1, f_2 \in F$ such that $f_1 1 + f_2 a = 0$, it follows that $f_2 a = -f_1$. If $f_2 \neq 0$, then $a = - \frac{f_1}{f_2} \in F$, a contradiction. Otherwise, we have $f_1 = 0$ as well, i.e. the only solution is $f_1 = f_2 = 0$. Thus $1, a$ are $F$-linearly independent.
Of course, there cannot exist two $F$-linearly independent elements in $K$ if $[K:F]=1$.
Edit: Sorry for the change in notation, I assumed $F \subset K$.
