In model theory, the finite spectrum of a first-order sentence $\phi $ (in a language with arbitrarily many constants, functions and relations is defined as the set of natural numbers $ n$ such that $\phi $ has a model with exactly $ n $ elements.
How can I show that the set of primes is a spectrum for a sentence $\phi $?
My first idea was to take the theory of fields and augment it by some symbols and theorems excluding the fields of order $ p^n $ with $ n > 1$, but I have no idea how to do that.