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2018 VNTST Problem 3

Source: 2018 Vietnam Team Selection Test

March 30, 2018
polynomialalgebra

Problem Statement

For every positive integer n≥3n\ge 3, let ϕn\phi_n be the set of all positive integers less than and coprime to nn. Consider the polynomial: Pn(x)=∑k∈ϕnxk−1.P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.
a. Prove that Pn(x)=(xrn+1)Qn(x)P_n(x)=(x^{r_n}+1)Q_n(x) for some positive integer rnr_n and polynomial Qn(x)∈Z[x]Q_n(x)\in\mathbb{Z}[x] (not necessary non-constant polynomial). b. Find all nn such that Pn(x)P_n(x) is irreducible over Z[x]\mathbb{Z}[x].
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