$[\mathbb{F}:\mathbb{K}]=1$ if and only if $\mathbb{F}=\mathbb{K}$.

Question

$\mathbb{F}$ is field extension of $\mathbb{K}$. I have to show $[\mathbb{F}:\mathbb{K}]=1$ if and only if $\mathbb{F}=\mathbb{K}$.

Answer 1

If $[\Bbb F : \Bbb K] = 1$, then as a vector space over $\Bbb K$, a basis for $\Bbb F$ has only one element. But $1$ is always an element of the basis. So $\{1 \}$ is the basis for $\Bbb F$ as a $\Bbb K$-vector space.

But if this is true, then each element in $\Bbb F$ can be written as $1 \cdot k$ for some $k \in \Bbb K$. But then that means every element of $\Bbb F$ is in $\Bbb K$, so $\Bbb F \subseteq \Bbb K$. And we know $\Bbb K \subseteq \Bbb F$, so we get $\Bbb F = \Bbb K$, as desired.