Many descendants

Algebra Level pending

Let \(p\) is a prime number. Denote by \(I(p)\) the number of irreducible monic polynomials with coefficients in \({\mathbb Z/pZ}\) (integers modulo \(p\)) that divide the polynomial \(x^{p+1}-x+1.\)

For \(p<100?\), what is the largest possible value of \(I(p)\) ?

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial \( x^3 + 3x - 5 \) is monic but the polynomial \( -x^4 + 2x^3 - 6 \) is not.

A polynomial with coefficients in \({\mathbb Z/pZ}\) is irreducible if it cannot be expressed as a product of two non-constant polynomials with coefficients in \({\mathbb Z/pZ}.\)


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