On computing subfields. A detailed description of the algorithm

Klüners, Jürgen

Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 243-271.

Soit $\mathbb{Q}\left(\alpha \right)$ un corps de nombres défini par le polynôme minimal de $\alpha$. Nous nous intéressons à déterminer les sous-corps $\mathbb{Q}\left(\beta \right)\subset \mathbb{Q}\left(\alpha \right)$ de degré donné. Chaque sous-corps est décrit en donnant le polynôme minimal $g$ de $\beta$ et le plongement de $\beta$ dans $\mathbb{Q}\left(\alpha \right)$ donné par un polynôme $h$ tel que $h\left(\alpha \right)=\beta$. Il y a une bijection entre les systèmes de blocs du groupe de Galois de $f$ et les sous-corps de $\mathbb{Q}\left(\alpha \right)$. 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.

Let $\mathbb{Q}\left(\alpha \right)$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha$. We want to determine all subfields $\mathbb{Q}\left(\beta \right)\subset \mathbb{Q}\left(\alpha \right)$ of given degree. It is convenient to describe each subfield by a pair $(g,h)\in \mathbb{Z}\left[t\right]\times \mathbb{Q}\left[t\right]$ such that $g$ is the minimal polynomial of $\beta =h\left(\alpha \right)$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $\mathbb{Q}\left(\alpha \right)$. 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.

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

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

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

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

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

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

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

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

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

[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

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

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

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

[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 | Zbl

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

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

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

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

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