I have shown that $F\subseteq K$.
I am trying to use the fact that K is a one-dimensional vector space over $F$ to show that $K\subseteq F$. I know that since $K$ is one-dimensional, it has one basis element, say $b$. So any $k \in K$ can be written as $k=fb$ where $f \in F$. Since we are dealing with fields, $f=\frac{k}{b}$. Does this show that $K\subseteq F$? I was told that we can just choose 1 as our basis. But why are we permitted to do this? Any guidance would be much appreciated.
Here you take $\{b\}\subset K$, which is basis of $K$ over $F$. And By definition a basis is linearly independent set. so here $b\neq 0_K$ must(where $0_K\in K$ is additive identity of $K$).
Now for any $a\in K$ , we have
$$\Rightarrow a.b\in K $$ $$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow \exists ~ f\in F ~\text{such that}~~ a\cdot b=f\cdot b $$ $$~~~~~~~~~~~~~~~~~\Rightarrow a=f\cdot b\cdot b^{-1}=f\in F$$ $$ \Rightarrow K\subset F$$