Computing local Artin maps, and solvability of norm equations
Based on the efficient computation of the automorphism group of abelian extensions of number fields by the authors [Math. Comput. 68, 1179-1186 (1999; Zbl 0937.11062)] and \textit{J. Klüners} [Über die Berechnung von Automorphismen und Teilkörpern algebraischer Zahlkörper. Berlin: TU Berlin (1997; Zbl 0912.11059)] they apply the results to the task of computing local and global Artin symbols. As an application they use Hasse's norm theorem to check for solvability of norm equations in cyclic extensions. In the previous works the authors computed the automorphism group of abelian extensions by explicitly constructing Frobenius automorphisms corresponding to certain unramified primes. Once the full group of automorphisms is known, it is straightforward to identify the Frobenius automorphism for any unramified prime ideal. Using some class field theory, the authors show how this knowledge of the global Artin correspondence enables them to compute arbitrary (ramified) local Artin maps. In order to apply this to the solvability of norm equations in cyclic extensions, they utilise Hasse's norm theorem stating that solvability of the norm equation is equivalent to the local solvability which is equivalent to the vanishing of the local Artin symbols. As a reduction step they show that only a finite number of local symbols have to be considered. They demonstrate the efficiency in several examples using Kash [KANT V4. J. Symb. Comput. 24, 267-283 (1997; Zbl 0886.11070)]. Since the previously implemented norm equation solver is particularly slow in the case on unsolvable equations, this greatly enhances the performance.
scientific article; zbMATH DE number 1532921