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). 




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$^1$ I made up a term for it because the only name I know for these is the rather unwieldy "[subterminal object](https://ncatlab.org/nlab/show/subterminal+object) in the opposite of the category of fields".



Is a field that never embeds twice in another field necessarily a prime field?
It seems that indeed only prime fields are unrepeatable.


**Proof:**
Let $k$ be unrepeatable and $F\subseteq k$ denote the prime field of $k$. Let $T\subseteq k$ be a transcendence base of $k/F$ and let $G=F(T)$. If $T\neq\emptyset$, then $G/F$ has non-trivial automorphisms (say take one element $t\in T$ to $t+1$). Since $k/G$ is algebraic this extends to a different embedding of $k$ into $\overline G$. Therefore, we see that $k/F$ must be algebraic. 


It follows that $F\subseteq k\subseteq \overline{F}$ and since $k$ is unrepeatable (in $\overline F$), $k$ has to be fixed by all automorphisms in the absolute Galois group of $F$. But then $F=k$ and we are done.


