no. 1
Computing all monogeneous mixed dihedral quartic extensions of a quadratic field
Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 137-142.

Soit M un corps quadratique réel. Nous donnons un algorithme rapide pour déterminer tous les corps quartiques diédraux K avec signature mixte, monogènes (i.e. ayant des bases d’entiers 1,α,α 2 ,α 3 ) et contenant M comme sous-corps. Nous déterminons également tous les générateurs α des bases dans K ayant cette forme. Notre algorithme combine un résultat récent de Kable [9] avec l’algorithme de Gaál, de Pethö et de Pohst [6], [7]. On applique la méthode à M=ℚ(2),ℚ(3),ℚ(5).

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).

@article{JTNB_2001__13_1_137_0,
     author = {Ga\'al, Istv\'an and Nyul, G\'abor},
     title = {Computing all monogeneous mixed dihedral quartic extensions of a quadratic field},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {137--142},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     mrnumber = {1838076},
     zbl = {1065.11086},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_2001__13_1_137_0/}
}
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
SP  - 137
EP  - 142
VL  - 13
IS  - 1
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_2001__13_1_137_0/
LA  - en
ID  - JTNB_2001__13_1_137_0
ER  - 
%0 Journal Article
%A Gaál, István
%A Nyul, Gábor
%T Computing all monogeneous mixed dihedral quartic extensions of a quadratic field
%J Journal de théorie des nombres de Bordeaux
%D 2001
%P 137-142
%V 13
%N 1
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_2001__13_1_137_0/
%G en
%F JTNB_2001__13_1_137_0
Gaál, István; Nyul, Gábor. Computing all monogeneous mixed dihedral quartic extensions of a quadratic field. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 137-142. http://archive.numdam.org/item/JTNB_2001__13_1_137_0/

[1] Y. Bilu, G. Hanrot, Solving Thue equations of high degree. J. Number Theory 60 (1996), 373-392. | MR | Zbl

[2] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, K. Wildanger, KANT V4. J. Symbolic Comp. 24 (1997), 267-283. | MR | Zbl

[3] I. Gaál, A. Pethö, M. Pohst, On the resolution of index form equations in biquadratic number fields, I. J. Number Theory 38 (1991), 18-34. | MR | Zbl

[4] I. Gaál, A. Pethö, M. Pohst, On the resolution of index form equations in biquadratic number fields, II. J. Number Theory 38 (1991), 35-51. | Zbl

[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 Theory 53 (1995), 100-114. | MR | Zbl

[6] I. Gaál, A. Pethö, M. Pohst, On the resolution of index form equations in quartic number fields. J. Symbolic Computation 16 (1993), 563-584. | MR | Zbl

[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 Theory 57 (1996), 90-104. | MR | Zbl

[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. | MR | Zbl

[9] A.C. Kable, Power integral bases in dihedral quartic fields. J. Number Theory 76 (1999), 120-129. | MR | Zbl

[10] L.C. Kappe, B. Warren, An elementary test for the Galois group of a quartic polynomial. Amer. Math. Monthly 96 (1989), 133-137. | MR | Zbl