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square residues

Source: Romanian IMO Team Selection Test TST 2004, problem 18

May 26, 2004
modular arithmeticquadraticsnumber theory proposednumber theory

Problem Statement

Let pp be a prime number and f∈Z[X]f\in \mathbb{Z}[X] given by f(x)=ap−1xp−2+ap−2xp−3+⋯+a2x+a1, f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , where ai=(ip)a_i = \left( \tfrac ip\right) is the Legendre symbol of ii with respect to pp (i.e. ai=1a_i=1 if ip−12≡1(modp) i^{\frac {p-1}2} \equiv 1 \pmod p and ai=−1a_i=-1 otherwise, for all i=1,2,…,p−1i=1,2,\ldots,p-1).
a) Prove that f(x)f(x) is divisible with (x−1)(x-1), but not with (x−1)2(x-1)^2 iff p≡3(mod4)p \equiv 3 \pmod 4; b) Prove that if p≡5(mod8)p\equiv 5 \pmod 8 then f(x)f(x) is divisible with (x−1)2(x-1)^2 but not with (x−1)3(x-1)^3.
Sugested by Calin Popescu.
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