# Does every finite field have a subfield $\mathbb{Z}_p$?

It seems that in the answers for my exercises in the book, the book uses that every finite field, has a subfield $\mathbb{Z}_p$. Is this true?

They seem to use it in the answer for one exercise. But first we have two exercises that the third build on, I will post them here for completness.

Now comes the exercise I am really wondering about. I will post it with its solution:

But why can the say that the field E has a subfield $\mathbb{Z}_p$? Does all finite fields have this kind of subfield, or does it follow in some way from exercise 30 and 36?

• Hint. Look at the set $\{1, 1+1, 1+1+1, ...\}$. Apr 24, 2015 at 18:15

Hint: Take the characteristic homomorphism $\rho: \mathbb Z \to F$, where $\rho (n) = n 1$.
Notice that $F$ is a domain and its characteristic must be a prime $p$ and $\rho (\mathbb Z) \simeq \mathbb Z_p$, because its kernel is $I(p)$, that is, the ideal generated by $p$ is the smallest subfield of $F$.
Yes it's true that every finite field contains $\mathbb F_p$ for one value of $p$. In fact every field contains either $\mathbb F_p$ for some $p$, or $\mathbb Q$.