On computing subfields. A detailed description of the algorithm

Jürgen Klüners

doi:10.5802/jtnb.227

Jürgen Klüners

Journal de Théorie des Nombres de Bordeaux, Volume 10 (1998) no. 2, pp. 243-271.

Abstract
Résumé

Let $ℚ (α)$ be an algebraic number field given by the minimal polynomial $f$ of $α$ . We want to determine all subfields $ℚ (β) \subset ℚ (α)$ of given degree. It is convenient to describe each subfield by a pair $(g, h) \in ℤ [t] \times ℚ [t]$ such that $g$ is the minimal polynomial of $β = h (α)$ . There is a bijection between the block systems of the Galois group of $f$ and the subfields of $ℚ (α)$ . These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding subfield using $p$ -adic methods. We give a detailed description for all parts of the algorithm.

Soit $ℚ (α)$ un corps de nombres défini par le polynôme minimal de $α$ . Nous nous intéressons à déterminer les sous-corps $ℚ (β) \subset ℚ (α)$ de degré donné. Chaque sous-corps est décrit en donnant le polynôme minimal $g$ de $β$ et le plongement de $β$ dans $ℚ (α)$ donné par un polynôme $h$ tel que $h (α) = β$ . Il y a une bijection entre les systèmes de blocs du groupe de Galois de $f$ et les sous-corps de $ℚ (α)$ . Ces systèmes de blocs sont calculés en utilisant les sous-groupes cycliques du groupe de Galois qui sont obtenus à partir du critère de Dedekind. Lorsqu’un système de blocs est connu, on calcule le sous-corps correspondants par des méthodes $p$ -adiques. Nous présentons ici une description détaillée de l’algorithme.

MR: 1828244 | Zbl: 0935.11047

DOI: 10.5802/jtnb.227

@article{JTNB_1998__10_2_243_0,
     author = {J\"urgen Kl\"uners},
     title = {On computing subfields. {A} detailed description of the algorithm},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {243--271},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {2},
     year = {1998},
     doi = {10.5802/jtnb.227},
     zbl = {0935.11047},
     mrnumber = {1828244},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.227/}
}

TY  - JOUR
TI  - On computing subfields. A detailed description of the algorithm
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 1998
DA  - 1998///
SP  - 243
EP  - 271
VL  - 10
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.227/
UR  - https://zbmath.org/?q=an%3A0935.11047
UR  - https://www.ams.org/mathscinet-getitem?mr=1828244
UR  - https://doi.org/10.5802/jtnb.227
DO  - 10.5802/jtnb.227
LA  - en
ID  - JTNB_1998__10_2_243_0
ER  -

%0 Journal Article
%T On computing subfields. A detailed description of the algorithm
%J Journal de Théorie des Nombres de Bordeaux
%D 1998
%P 243-271
%V 10
%N 2
%I Université Bordeaux I
%U https://doi.org/10.5802/jtnb.227
%R 10.5802/jtnb.227
%G en
%F JTNB_1998__10_2_243_0

Jürgen Klüners. On computing subfields. A detailed description of the algorithm. Journal de Théorie des Nombres de Bordeaux, Volume 10 (1998) no. 2, pp. 243-271. doi : 10.5802/jtnb.227. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.227/

References
Cited by

[1] D. Casperson, D. Ford, J. Mckay, Ideal decompositions and subfields. J. Symbolic Comput. 21 (1996), 133-137. | MR: 1394600 | Zbl: 0849.68071

[2] J.W.S. Cassels, Local Fields. Cambridge University Press, 1986. | MR: 861410 | Zbl: 0595.12006

[3] H. Cohen, F. Diaz Y Diaz, A polynomial reduction algorithm. Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 2, 351-360. | Numdam | MR: 1149802 | Zbl: 0758.11053

[4] Henri Cohen, A. Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, 1993. | MR: 1228206 | Zbl: 0786.11071

[5] G.E. Collins, M.E. Encarnación, Efficient rational number reconstruction. J. Symbolic Comput 20 (1995), 287-297. | MR: 1378101 | Zbl: 0851.68037

[6] Mario Daberkow, Claus Fieker, Jürgen Klüners, Michael Pohst, Katherine Roegner, Klaus Wildanger, KANT V4. J. Symbolic Comput. 24 (1997), 267-283. | MR: 1484479 | Zbl: 0886.11070

[7] F. Diaz Y Diaz, M. Olivier, Imprimitive ninth-degree number fields with small discriminants. Math. Comput. 64 (1995), no. 209, 305-321. | MR: 1260128 | Zbl: 0819.11070

[8] J. Dixon, Computing subfields in algebraic number fields. J. Austral. Math. Soc. Ser. A 49 (1990), 434-448. | MR: 1074513 | Zbl: 0727.11049

[9] A. Hulpke, Block systems of a Galois group. Experiment. Math. 4 (1995), no. 1, 1-9. | MR: 1359413 | Zbl: 0976.12006

[10] J. Klüners, Über die Berechnung von Teilkörpern algebraischer Zahlkörper. Diplomarbeit, Technische Universität Berlin, 1995.

[11] J. Klilners, Über die Berechnung von Automorphismen und Teilkörpern algebraischer Zahlkörper. Dissertation, Technische Universität Berlin, 1997. | Zbl: 0912.11059

[12] J. Klüners M. Pohst, On computing subfields. J. Symbolic Comput. 24 (1997), 385-397. | MR: 1484487 | Zbl: 0886.11072

[13] S. Landau, Factoring polynomials over algebraic number fields. SIAM J. Comput. 14 (1985), no. 1, 184-195. | MR: 774938 | Zbl: 0565.12002

[14] S. Landau, G.L. Miller, Solvability by radicals is in polynomial time. J. Comput. System Sci. 30 (1985), no. 2, 179-208. | MR: 801822 | Zbl: 0586.12002

[15] D. Lazard, A. Valibouze, Computing subfields: Reverse of the primitive element problem. In A. Galligo F. Eyssete, editor, MEGA-92, Computational algebraic geometry, volume 109, pages 163-176. Birkhäuser, Boston, 1993. | MR: 1230864 | Zbl: 0798.12002

[16] M. Mignotte, An inequality about factors of polynomials. Math. Comput. 28 (1974), no. 128, 1153-1157. | MR: 354624 | Zbl: 0299.12101

[17] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers. Springer-Verlag, 1990. | MR: 1055830 | Zbl: 02107000

[18] M.E. Pohst, H. Zassenhaus, Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications, 30. Cambridge University Press, Cambridge 1989 | MR: 1033013 | Zbl: 0685.12001

[19] P.J. Weinberger, L. Rothschild, Factoring polynomials over algebraic number fields. ACM Trans. Math. Software 2 (1976), no. 4, 335-350. | MR: 450225 | Zbl: 0352.12003

[20] H. Wielandt, Finite Permutation Groups. Academic Press, New York-London 1964. | MR: 183775 | Zbl: 0138.02501

Cited by Sources: