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Solvability of norm equations over cyclic number fields of prime degree

Authors:	Vincenzo Acciaro

Institution:	School of Computer Science, Carleton University, Ottawa, Ontario, K1S 5B6, Canada

Abstract:	Let $L={\mathbb {Q}} \alpha ]$ be an abelian number field of prime degree , and let be a nonzero rational number. We describe an algorithm which takes as input and the minimal polynomial of $\alpha$ over ${\mathbb {Q}}$ , and determines if is a norm of an element of . We show that, if we ignore the time needed to obtain a complete factorization of and a complete factorization of the discriminant of $\alpha$ , 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 $A=( E, \sigma , a )$ over ${\mathbb {Q}}$ is a division algebra.

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