id: 01532921
dt: j
an: 01532921
au: Acciaro, Vincenzo; Klüners, Jürgen
ti: Computing local Artin maps, and solvability of norm equations.
so: J. Symb. Comput. 30, No.3, 239-252 (2000).
py: 2000
pu: Elsevier Science (Academic Press), London
la: EN
cc: 
ut: automorphism group; Artin symbol; Frobenius automorphism; norm equations;
    Hasse norm theorem
ci: Zbl 0937.11062; Zbl 0912.11059; Zbl 0886.11070
li: doi:10.1006/jsco.2000.0361
ab: 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 {\it 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.
rv: Claus Fieker (Berlin)