Problem: Suppose are primes and is a positive integer so that , , and . Let be integers such that is a multiple of . Prove that at least one of the ‘s is a multiple of . [Problems from the Book, 23.14]
Solution: The nonzero are just the th roots of unity modulo . Let the th roots of unity be . We want to show that if for nonnegative integers then either (in which case ) or . Assume the latter is not true.
Let be a primitive root of unity (in ). The Galois group of consists of the maps for . By the Fixed Field Theorem the fixed field is . Hence
(Alternatively, just see that the coefficients will be symmetric polynomials of the .) Now
Considering the polynomial modulo , we see that is a root (replace the th roots of unity in by the th roots of unity modulo . This is okay since the th roots of unity modulo satisfy every algebraic relation over that the th roots of unity in do.). Since it is zero modulo , the constant term must be zero modulo , and (since its absolute value is less than ). This means that one of the factors is zero and all the are equal (since is the irreducible polynomial of ).