# $\mathbb Q$ and $\mathbb Z_p$ are prime fields

A field is called a prime field if it has no proper subfields.

How do I show $\mathbb Q$ and $\mathbb Z_p$ are prime fields? P is a prime.

Intuitively, it is clearly since any subfield of the above two fields must contain 1 and if it contains 1 it must contain every thing.

But how do I put it rigorously?

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## 1 Answer

Recall that for any (unital) ring $R$, there is exactly one (unital) ring homomorphism $f:\mathbb{Z}\to R$, namely the one defined by $$f(n)=\begin{cases}\underbrace{1_R+\cdots+1_R}_{n\text{ times}}\text{ if }n\geq0\\\\\\\\\\ -\underbrace{(1_R+\cdots+1_R)}_{-n\text{ times}}\text{ if }n<0\end{cases}$$

Now, you're right that the idea here is that any subfield must contain $1$, and then we should look at the subfield "generated" by 1. Here is one way of making this precise: suppose $K\subseteq \mathbb{Q}$ is a field. Consider the image of the unique ring homomorphism $f:\mathbb{Z}\to K$. What must be the kernel of $f$? Can you prove that if $\mathbb{Z}\subseteq K$, then $\mathbb{Q}\subseteq K$?

Similarly, suppose $K\subseteq\mathbb{Z}_p$ is a field. What can possibly occur as the kernel of $f:\mathbb{Z}\to K$?

This is closely related to my answer here.

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