Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

István Gaál; Gábor Nyul

István Gaál ; Gábor Nyul

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 137-142.

Résumé
Abstract

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 $M = ℚ (\sqrt{2}), ℚ (\sqrt{3}), ℚ (\sqrt{5}) .$

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 $M = ℚ (\sqrt{2}), ℚ (\sqrt{3}), ℚ (\sqrt{5}) .$

MR Zbl

@article{JTNB_2001__13_1_137_0,
     author = {Istv\'an Ga\'al and G\'abor Nyul},
     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},
     zbl = {1065.11086},
     mrnumber = {1838076},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_137_0/}
}

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István Gaál; Gábor Nyul. 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. https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_137_0/

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