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Mathematics of Computation

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Solving norm equations in relative number fields using $S$-units

Author: Denis Simon
Journal: Math. Comp. 71 (2002), 1287-1305
MSC (2000): Primary 11D57, 11Y50, 11R29
Published electronically: January 11, 2002
MathSciNet review: 1898758
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Abstract: In this paper, we are interested in solving the so-called norm equation ${\mathcal N}_{L/K} (x)=a$, where $L/K$ is a given arbitrary extension of number fields and $a$ a given algebraic number of $K$. By considering $S$-units and relative class groups, we show that if there exists at least one solution (in $L$, but not necessarily in ${\mathbb Z}_L$), then there exists a solution for which we can describe precisely its prime ideal factorization. In fact, we prove that under some explicit conditions, the $S$-units that are norms are norms of $S$-units. This allows us to limit the search for rational solutions to a finite number of tests, and we give the corresponding algorithm. When $a$ is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.

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Additional Information

Denis Simon
Affiliation: Université Bordeaux I, Laboratoire A2X, 351 Cours de la Libération, 33405 Talence, France

Keywords: Relative number fields, norm equation, $S$-unit, class group
Received by editor(s): January 22, 1999
Received by editor(s) in revised form: April 13, 1999
Published electronically: January 11, 2002
Article copyright: © Copyright 2002 American Mathematical Society