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Mathematics of Computation

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Solvability of norm equations over cyclic number fields of prime degree

Author: Vincenzo Acciaro
Journal: Math. Comp. 65 (1996), 1663-1674
MSC (1991): Primary 11R37; Secondary 11Y40
MathSciNet review: 1351200
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Abstract: Let $L={\mathbb {Q}} [\alpha ]$ 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 $\alpha $ over ${\mathbb {Q}}$, and determines if $a$ is a norm of an element of $L$. We show that, if we ignore the time needed to obtain a complete factorization of $a$ 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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Vincenzo Acciaro
Affiliation: School of Computer Science, Carleton University, Ottawa, Ontario, K1S 5B6, Canada

Received by editor(s): March 30, 1995
Received by editor(s) in revised form: July 14, 1995
Article copyright: © Copyright 1996 American Mathematical Society