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square free mod n

Source: Iran TST2 Day1 P1

March 11, 2020
polynomialnumber theoryIranian TST

Problem Statement

We call a monic polynomial P(x)∈Z[x]P(x) \in \mathbb{Z}[x] square-free mod n if there dose not exist polynomials Q(x),R(x)∈Z[x]Q(x),R(x) \in \mathbb{Z}[x] with QQ being non-constant and P(x)≡Q(x)2R(x)mod  nP(x) \equiv Q(x)^2 R(x) \mod n. Given a prime pp and integer m≥2m \geq 2. Find the number of monic square-free mod p P(x)P(x) with degree mm and coeeficients in {0,1,2,3,...,p−1}\{0,1,2,3,...,p-1\}.
Proposed by Masud Shafaie
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