Let n=p1p2⋯ps, where p1,…,ps are distinct odd prime numbers.
(a) Prove that the expression
Fn(x)=∏(xpi1⋯pikn−1)(−1)k,where the product goes over all subsets {pi1,…,pik} or {p1,…,ps} (including itself and the empty set), can be written as a polynomial in x with integer coefficients.
(b) Prove that if p is a prime divisor of Fn(2), then either p∣n or n∣p−1.
functionPolynomialsnumber theory