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Let $F$ be a field with $|F|=q$. Determine with proof the number of monic irreducible polynomials of prime degree $p$ over $F$, where $p$ need not be the characteristic of $F$.

I know that $x^{p^n} -x$ factors over $\Bbb F_p$ into the product of all monic irreducible polynomials over $\Bbb F_p$ of degree a divisor of $n$. But how to solve the question?

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The way you’ve posed this, it looks like a homework problem, so I won’t give the full answer. An irreducible polynomial $f$ of degree $p$ has $p$ different roots, each of which lies in the field $K$ with $q^p$ elements. Every element of $K$ is either in $F$ or satisfies an irreducible polynomial over $F$ of degree $p$. You should be able to take it from there.

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