Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

István Gaál; Gábor Nyul

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 137-142
  • ISSN: 1246-7405

Abstract

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Let M be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields K with mixed signature having power integral bases and containing M as a subfield. We also determine all generators of power integral bases in K . Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for M = ℚ ( 2 ) , ℚ ( 3 ) , ℚ ( 5 ) .

How to cite

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Gaál, István, and Nyul, Gábor. "Computing all monogeneous mixed dihedral quartic extensions of a quadratic field." Journal de théorie des nombres de Bordeaux 13.1 (2001): 137-142. <http://eudml.org/doc/248697>.

@article{Gaál2001,
abstract = {Let $M$ be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields $K$ with mixed signature having power integral bases and containing $M$ as a subfield. We also determine all generators of power integral bases in $K$. Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for $M = \mathbb \{Q\}(\sqrt\{2\}), \mathbb \{Q\}(\sqrt\{3\}), \mathbb \{Q\}(\sqrt\{5\}).$},
author = {Gaál, István, Nyul, Gábor},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {power integral basis; dihedral quartic field; quadratic subfield},
language = {eng},
number = {1},
pages = {137-142},
publisher = {Université Bordeaux I},
title = {Computing all monogeneous mixed dihedral quartic extensions of a quadratic field},
url = {http://eudml.org/doc/248697},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Gaál, István
AU - Nyul, Gábor
TI - Computing all monogeneous mixed dihedral quartic extensions of a quadratic field
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 137
EP - 142
AB - Let $M$ be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields $K$ with mixed signature having power integral bases and containing $M$ as a subfield. We also determine all generators of power integral bases in $K$. Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for $M = \mathbb {Q}(\sqrt{2}), \mathbb {Q}(\sqrt{3}), \mathbb {Q}(\sqrt{5}).$
LA - eng
KW - power integral basis; dihedral quartic field; quadratic subfield
UR - http://eudml.org/doc/248697
ER -

References

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  1. [1] Y. Bilu, G. Hanrot, Solving Thue equations of high degree. J. Number Theory60 (1996), 373-392. Zbl0867.11017MR1412969
  2. [2] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, K. Wildanger, KANT V4. J. Symbolic Comp.24 (1997), 267-283. Zbl0886.11070MR1484479
  3. [3] I. Gaál, A. Pethö, M. Pohst, On the resolution of index form equations in biquadratic number fields, I. J. Number Theory38 (1991), 18-34. Zbl0726.11022MR1105669
  4. [4] I. Gaál, A. Pethö, M. Pohst, On the resolution of index form equations in biquadratic number fields, II. J. Number Theory38 (1991), 35-51. Zbl0726.11023
  5. [5] I. Gaál, A. Pethö, M. Pohst, On the resolution of index form equations in biquadratic number fields, III. The bicyclic biquadratic case. J. Number Theory53 (1995), 100-114. Zbl0853.11026MR1344834
  6. [6] I. Gaál, A. Pethö, M. Pohst, On the resolution of index form equations in quartic number fields. J. Symbolic Computation16 (1993), 563-584. Zbl0808.11023MR1279534
  7. [7] I. Gaál, A. Pethö, M. Pohst, Simultaneous representation of integers by a pair of ternary quadratic forms - with an application to index form equations in quartic number fields. J. Number Theory57 (1996), 90-104. Zbl0853.11023MR1378574
  8. [8] I. Gaál, A. Pethö, M. Pohst, On the resolution of index form equations in dihedral number fields. J. Experimental Math.3 (1994), 245-254. Zbl0823.11074MR1329372
  9. [9] A.C. Kable, Power integral bases in dihedral quartic fields. J. Number Theory76 (1999), 120-129. Zbl0934.11051MR1688180
  10. [10] L.C. Kappe, B. Warren, An elementary test for the Galois group of a quartic polynomial. Amer. Math. Monthly96 (1989), 133-137. Zbl0702.11075MR992075

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