Show that any field K has a subfield isomorphic to either $\mathbb{Q}$ of $\mathbb{Z}_p$
I understand that here we are talking about a prime subfield that would be isomorphic to either one or the other.
My attempt/understanding:
Considering the characteristic of the field, we know it must be either zero or some prime number (since there are no other possibilities). If the characteristic is zero, then the field is infinite, like $\mathbb{Q}(\sqrt{3})$ or $\mathbb{C}$ and clearly the prime subfield would be $\mathbb{Q}$. If my characteristic is finite, then it must be some prime, then my field is something like $\mathbb{Z}_2/<x^2+x+1>$, and the underlying prime subfield should obviously be $\mathbb{Z}_2$. I guess I'm not sure how to set up a proper proof, but I know that the characteristic is going to be a big part of it.
Best Answer
The prime subfield will be the subfield obtained by taking the additive subgroup generated by $1$ and then throwing in multiplicative inverses.
If $K$ has characteristic $0$, then the additive subgroup generated by $1$ will be isormorphic to $\mathbb{Z}$ and so adding inverses gives that the prime subfield is isomorphic to $\mathbb{Q}$.
If $K$ has characteristic $p$, then the additive subgroup generated by $1$ is isormophic to $\mathbb{Z}_p$ as an additive group. The non-zero elements of $\mathbb{Z}_p$ have multiplicative inverses, so the prime subfield is isomorphic to $\mathbb{Z}_p$.