Irreducible Polynomials High

Let $p$ be a prime number. Consider \mathbf{Z}_p = \{0, 1, \ldots, p-1\} — the field of residues modulo $p$. In this field, addition and multiplication operations are performed modulo $p$.

Now consider monic polynomials over this field of the form

where $n$ is the degree of the polynomial, $x$ is the variable, and the coefficients a_i belong to \mathbf{Z}_p.

For example, the polynomial x^2 + x + 1 over the field \mathbf{Z}_2 is irreducible. A monic polynomial is called irreducible if it cannot be written as f(x) = p(x) \cdot q(x), where p(x), q(x) are polynomials of degree less than that of $f(x)$. And it is the only irreducible degree 2 monic polynomial over the field \mathbf{Z}_2.

Your task is to compute the number of irreducible monic polynomials of degree $n$ over the field \mathbf{Z}_p. Since this number can be large, you need to find the remainder of this number when divided by $m$.