Normal integral bases for $A_4$ extensions of the rationals
HTML articles powered by AMS MathViewer
- by Jean Cougnard PDF
- Math. Comp. 75 (2006), 485-496 Request permission
Abstract:
We give an algorithm for constructing normal integral bases of tame Galois extensions of the rationals with group $A_4$. Using earlier works we can do the same until degree $15$.References
- E. Artin, Questions de base minimale dans la théorie des nombres algébriques, Algèbre et Théorie des Nombres., Colloques Internationaux du Centre National de la Recherche Scientifique, no. 24, Centre National de la Recherche Scientifique, Paris, 1950, pp. 19–20 (French). MR 0042450
- Albert Châtelet, Arithmétique des corps abéliens du troisième degré, Ann. Sci. École Norm. Sup. (3) 63 (1946), 109–160 (1947) (French). MR 0020598, DOI 10.24033/asens.934
- Jean Cougnard, Anneaux d’entiers stablement libres sur $\Bbb Z[H_8\times C_2]$, J. Théor. Nombres Bordeaux 10 (1998), no. 1, 163–201 (French, with English and French summaries). MR 1827291, DOI 10.5802/jtnb.224
- Jean Cougnard, Construction de base normale pour les extensions de $\textbf {Q}$ à groupe $D_4$, J. Théor. Nombres Bordeaux 12 (2000), no. 2, 399–409 (French, with English and French summaries). Colloque International de Théorie des Nombres (Talence, 1999). MR 1823192, DOI 10.5802/jtnb.286
- Jean Cougnard and Jacques Queyrut, Construction de bases normales pour les extensions galoisiennes absolues à groupe de Galois quaternionien d’ordre 12, J. Théor. Nombres Bordeaux 14 (2002), no. 1, 87–102 (French, with English and French summaries). MR 1925992, DOI 10.5802/jtnb.348
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
- Jacques Martinet, Sur l’arithmétique des extensions galoisiennes à groupe de Galois diédral d’ordre $2p$, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 1–80, ix (French, with English summary). MR 262210, DOI 10.5802/aif.307
- C. BATUT, K. BELABAS, D. BERNARDI, H. COHEN, and M. OLIVIER. User’s Guide to Pari-GP, version 2.02.12 (1999).
- Jean-Jacques Payan, Critère de décomposition d’une extension de Kummer sur un sous-corps du corps de base, Ann. Sci. École Norm. Sup. (4) 1 (1968), 445–458 (French). MR 237472, DOI 10.24033/asens.1169
- Dock Sang Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700–712. MR 104721, DOI 10.2307/1970033
- I. Reiner and S. Ullom, Remarks on class groups of integral group rings, Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 501–516. MR 0367043
Additional Information
- Received by editor(s): March 28, 2004
- Received by editor(s) in revised form: October 28, 2004
- Published electronically: September 1, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 75 (2006), 485-496
- MSC (2000): Primary 11R04, 11Y40; Secondary 11R33
- DOI: https://doi.org/10.1090/S0025-5718-05-01779-5
- MathSciNet review: 2176411