The characteristic of a field is the smallest positive integer $n$ such that $n\times 1=0$, or $\infty$ if there is no such integer. If it is not infinite, then it has to be a prime number.
A couple of relevant results:
- $\Bbb F_p$ has characteristic $p$ for any prime $p$.
- If $K$ is a subfield of $L$, then they both have the same characteristic.
As a consequence, there can be no field containing $\Bbb F_5$ and $\Bbb F_7$ as subfields.
Edit: $\Bbb F_p$ here is just the field notation for $\Bbb Z_p$, when you want to emphasize that you are talking about the field rather than the group.
Edit to include the answers to your questions in comments:
Proof of 2. above. Call $\phi$ the injective field homomorphism $K\to L$. Then $\phi(n\times 1_K)=n\times 1_L$ for any $n\in \Bbb N$. This implies, on the one hand, that if $n\times 1_K=0$ then $n\times 1_L=0$. On the other hand, since $\phi$ is injective, it also implies that if $n\times 1_L=0$ then $n\times 1_K=0$.
An infinite field may have finite characteristic. The field of fractions of $\Bbb Z_2[X]$ has characteristic $2$, but it is infinite.