SOLVABILITY OF NORM EQUATIONS OVER CYCLIC NUMBER-FIELDS OF PRIME DEGREE

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