Solvability of norm equations over cyclic number fields of prime degree

Abstract

Let L = Q[α] be an abelian number field of prime degree q, and let α be a nonzero rational number. We describe an algorithm which takes as input α and the minimal polynomial of α over Q, and determines if α is a norm of an element of L. We show that, if we ignore the time needed to obtain a complete factorization of α and a complete factorization of the discriminant of α, then the algorithm runs in time polynomial in the size of the input. As an application, we give an algorithm to test if a cyclic algebra A = (E, σ, σ) over Q is a division algebra.