RO-TST 2004 Problem 18

Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by $$f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 ,$$ where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$ ).

a) Prove that $f(x)$ is divisible with $(x-1)$ , but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$ ; b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$ .

Sugested by Calin Popescu.