the definition of a prime subfield that I have been provided with is the following:
It can be shown that for any subset $S$ of $F$ there exists a minimal subfield $K$ of $F$ which contains $S$. One very important case is when the set $S$ is empty, i.e., every field $F$ contains a unique minimal subfield. This subfield is called the prime subfield.
I further know that any subfield $K$ of a field $F$ contains the elements $0$ and $1$ and is closed under addition, multiplication, and taking negatives and inverses of non-zero elements.
And since the prime subfield is the intersection of all possible subfields across $F$, why is it not the case that the prime subfield is always equal to $0$ and $1$?
Thank you for the help!