Call a field $k$ *unrepeatable*$^1$ if for every field $L$ there are either zero or one field homomorphisms $k \to L$. Then the prime fields $\mathbb{Q}$ and $\mathbb{F}_p$ for $p$ prime are clearly unrepeatable and it seems very likely to me that those are the only ones. Is that true?

Notice that an unrepeatable field cannot have a non-identity automorphism, and must have a unique embedding into its algebraic closure (so for example $\mathbb{R}$ is not unrepeatable despite having no nontrivial automorphisms).

$^1$ I made up a term for it because the only name I know for these is the rather unwieldy "subterminal object in the opposite of the category of fields".

endomorphism: it isn't always surjective. It's existence and non-identity-ness still shows unrepeatable positive characteristic fields are prime as you said.) $\endgroup$ – Omar Antolín-Camarena Feb 28 '18 at 5:21