On computing subfields. A detailed description of the algorithm

Jürgen Klüners

Journal de théorie des nombres de Bordeaux (1998)

  • Volume: 10, Issue: 2, page 243-271
  • ISSN: 1246-7405

Abstract

top
Let ℚ ( α ) be an algebraic number field given by the minimal polynomial f of α . We want to determine all subfields ℚ ( β ) ⊂ ℚ ( α ) of given degree. It is convenient to describe each subfield by a pair ( g , h ) ∈ ℤ [ t ] × 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.

How to cite

top

Klüners, Jürgen. "On computing subfields. A detailed description of the algorithm." Journal de théorie des nombres de Bordeaux 10.2 (1998): 243-271. <http://eudml.org/doc/248167>.

@article{Klüners1998,
abstract = {Let $\mathbb \{Q\}\{(\alpha )\}$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha $. We want to determine all subfields $\mathbb \{Q\}\{(\beta )\} \subset \mathbb \{Q\}\{(\alpha )\}$ of given degree. It is convenient to describe each subfield by a pair $(g , h) \in \mathbb \{Z\} [t] \times \mathbb \{Q\}[t]$ such that $g$ is the minimal polynomial of $\beta = h(\alpha )$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $\mathbb \{Q\}\{(\alpha )\}$. 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.},
author = {Klüners, Jürgen},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {subfields; block systems},
language = {eng},
number = {2},
pages = {243-271},
publisher = {Université Bordeaux I},
title = {On computing subfields. A detailed description of the algorithm},
url = {http://eudml.org/doc/248167},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Klüners, Jürgen
TI - On computing subfields. A detailed description of the algorithm
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 2
SP - 243
EP - 271
AB - Let $\mathbb {Q}{(\alpha )}$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha $. We want to determine all subfields $\mathbb {Q}{(\beta )} \subset \mathbb {Q}{(\alpha )}$ of given degree. It is convenient to describe each subfield by a pair $(g , h) \in \mathbb {Z} [t] \times \mathbb {Q}[t]$ such that $g$ is the minimal polynomial of $\beta = h(\alpha )$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $\mathbb {Q}{(\alpha )}$. 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.
LA - eng
KW - subfields; block systems
UR - http://eudml.org/doc/248167
ER -

References

top
  1. [1] D. Casperson, D. Ford, J. McKay, Ideal decompositions and subfields. J. Symbolic Comput.21 (1996), 133-137. Zbl0849.68071MR1394600
  2. [2] J.W.S. Cassels, Local Fields. Cambridge University Press, 1986. Zbl0595.12006MR861410
  3. [3] H. Cohen, F. Diaz y Diaz, A polynomial reduction algorithm. Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 2, 351-360. Zbl0758.11053MR1149802
  4. [4] Henri Cohen, A.Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, 1993. Zbl0786.11071MR1228206
  5. [5] G.E. Collins, M.E. Encarnación, Efficient rational number reconstruction. J. Symbolic Comput20 (1995), 287-297. Zbl0851.68037MR1378101
  6. [6] Mario Daberkow, Claus Fieker, Jürgen Klüners, Michael Pohst, Katherine Roegner, Klaus Wildanger, KANT V4. J. Symbolic Comput. 24 (1997), 267-283. Zbl0886.11070MR1484479
  7. [7] F. Diaz y Diaz, M. Olivier, Imprimitive ninth-degree number fields with small discriminants. Math. Comput.64 (1995), no. 209, 305-321. Zbl0819.11070MR1260128
  8. [8] J. Dixon, Computing subfields in algebraic number fields. J. Austral. Math. Soc. Ser. A49 (1990), 434-448. Zbl0727.11049MR1074513
  9. [9] A. Hulpke, Block systems of a Galois group. Experiment. Math.4 (1995), no. 1, 1-9. Zbl0976.12006MR1359413
  10. [10] J. Klüners, Über die Berechnung von Teilkörpern algebraischer Zahlkörper. Diplomarbeit, Technische UniversitätBerlin, 1995. 
  11. [11] J. Klilners, Über die Berechnung von Automorphismen und Teilkörpern algebraischer Zahlkörper. Dissertation, Technische UniversitätBerlin, 1997. Zbl0912.11059
  12. [12] J. Klüners M. Pohst, On computing subfields. J. Symbolic Comput.24 (1997), 385-397. Zbl0886.11072MR1484487
  13. [13] S. Landau, Factoring polynomials over algebraic number fields. SIAM J. Comput.14 (1985), no. 1, 184-195. Zbl0565.12002MR774938
  14. [14] S. Landau, G.L. Miller, Solvability by radicals is in polynomial time. J. Comput. System Sci.30 (1985), no. 2, 179-208. Zbl0586.12002MR801822
  15. [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. Zbl0798.12002MR1230864
  16. [16] M. Mignotte, An inequality about factors of polynomials. Math. Comput.28 (1974), no. 128, 1153-1157. Zbl0299.12101MR354624
  17. [17] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers. Springer-Verlag, 1990. Zbl1159.11039MR1055830
  18. [18] M.E. Pohst, H. Zassenhaus, Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications, 30. Cambridge University Press, Cambridge1989 Zbl0685.12001MR1033013
  19. [19] P.J. Weinberger, L. Rothschild, Factoring polynomials over algebraic number fields. ACM Trans. Math. Software2 (1976), no. 4, 335-350. Zbl0352.12003MR450225
  20. [20] H. Wielandt, Finite Permutation Groups. Academic Press, New York-London1964. Zbl0138.02501MR183775

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.