I am currently trying to prove that, given that $K$ is a field and $k\subset K$ is its prime-field, that \begin{equation} k =\bigcap_{\small{\overline{K} \text{ subfield of } K}}\overline{K}. \end{equation}
Here are my definitions:
If the characteristic of $K$ is $0$: There is a unique $f:\mathbb{Q}\rightarrow K$ which is injective.
If the characteristic of $K$ is $p$: There is a unique $f:\mathbb{Z}/ p\mathbb{Z}\ \rightarrow K$ which is injective.
The prime subfield is defined as Im$(f)$, which I think would mean that $\mathbb{Q}\cong k$ or $\mathbb{Z}/ p\mathbb{Z}\cong k$ (depending on the characteristic of $K$).
Now to my question: Do I need to look at both cases and construct the smallest possible subfield of $K$? In the first case this would be $\mathbb{Q}$ because every field needs to contain {$0,1$} which we can build up to $\mathbb{Q}$. Where do I need the isomorphism from $k$ to $\mathbb{Z}/ p\mathbb{Z}$ or $\mathbb{Q}$ ? I am having troubles with my intuition when it comes to algebra, so any push into the right direction would be appreciated.