A first order theory whose finite models are exactly the $\Bbb F_p$

Question

Is there a first order theory $T$ in the language of rings such that its finite models are exactly the fields $\Bbb F_p$ with $p$ prime (but no $\Bbb F_q$ with $q$ a proper prime power is a model of $T$)?

EDIT: Since this question turned out to be trivial, I asked if it is also possible with a finite theory. See here

Answer 1

Let $T$ be the theory of fields with the additional schema:

If $p\cdot1=0$ then for every $x$ it holds $x=0$ or $x=1$ or $x=1+1$ or ... $x=(p-1)\cdot 1$.

Now if $F$ is a finite model of $T$ then $F$ is a field, and it has exactly the same number of elements as its characteristics.

Note that $F$ is an infinite model of $T$ if and only if it is a field of characteristics zero.