On computing subfields. A detailed description of the algorithm
Journal de théorie des nombres de Bordeaux, Volume 10 (1998) no. 2, p. 243-271

Let (α) be an algebraic number field given by the minimal polynomial f of α. We want to determine all subfields (β)(α) of given degree. It is convenient to describe each subfield by a pair (g,h)[t]×[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 (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 (α) un corps de nombres défini par le polynôme minimal de α. Nous nous intéressons à déterminer les sous-corps (β)(α) de degré donné. Chaque sous-corps est décrit en donnant le polynôme minimal g de β et le plongement de β dans (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 (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.

@article{JTNB_1998__10_2_243_0,
     author = {Kl\"uners, J\"urgen},
     title = {On computing subfields. A detailed description of the algorithm},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {2},
     year = {1998},
     pages = {243-271},
     zbl = {0935.11047},
     mrnumber = {1828244},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1998__10_2_243_0}
}
Klüners, Jürgen. On computing subfields. A detailed description of the algorithm. Journal de théorie des nombres de Bordeaux, Volume 10 (1998) no. 2, pp. 243-271. http://www.numdam.org/item/JTNB_1998__10_2_243_0/

[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