$\mathbb Q$ is the only infinite prime field

Question

I'd like to show that

$\mathbb Q$ is the only infinite prime field (up to isomorphism of fields.)

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A prime field is a field that doesn't contain any proper subfields. It's clear that $\Bbb Q$ is a prime field. Suppose $K$ is an infinite prime field with identity element $1_K$. We'd like to construct a field isomorphism $\phi: K\to \Bbb Q$. Needless to say, $\phi(1_K) = 1$. How do we define the action of $\phi$ on an arbitrary $k\in K$? Perhaps this is a basic question - so just hints shall suffice too. Thank you!

Edit: I was looking to construct $\phi$ (see my attempt above), which is not answered by the linked question, hence this post is not a duplicate.

Answer 1

Hint: there's a straightforward field embedding $\Bbb Q\to K$ for any field of characteristic zero. If $K$ is prime, then considering that the image of $\Bbb Q$ should be a field too, you find the embedding surjects.

Is it possible for $K$ to have positive characteristic? Try constructing a finite subfield in this instance.