On the Multiplicity of Discriminants of Relative Quadratic Extensions of Quintic Fields

Authors: Schehrazad Selmane

Abstract:

According to Hermite there exists only a finite number of number fields having a given degree, and a given value of the discriminant, nevertheless this number is not known generally. The determination of a maximum number of number fields of degree 10 having a given discriminant that contain a subfield of degree 5 having a fixed class number, narrow class number and Galois group is the purpose of this work. The constructed lists of the first coincidences of 52 (resp. 50, 40, 48, 22, 6) nonisomorphic number fields with same discriminant of degree 10 of signature (6,2) (resp. (4,3), (8,1), (2,4), (0,5), (10,0)) containing a quintic field. For each field in the lists, we indicate its discriminant, the discriminant of its subfield, a relative polynomial generating the field over its quintic field and its relative discriminant, the corresponding polynomial over Q and its Galois closure are presented with concluding remarks.

Keywords: Discriminant, nonisomorphic fields, quintic fields, relative quadratic extensions.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1335590

References:

[1] G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Alg. 11 (1983), pp. 863–911.
[2] M. Daberkow, C. Fieker, J. Kluners, M. Pohst, K. Roegner and K. Wildanger: KANT V4, in J. Symbolic Comp. 24 (1997), pp. 267–283.
[3] H. Hasse, Arithmetische Theorie der kubischen Zahlk¨orper auf klassenk¨orpertheoretischer Grundlage, Math. Z. 31 (1930), pp. 565– 582.
[4] D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), no. 198, 831-847 and Supplements section S55–S58.
[5] D. C. Mayer, Discriminants of metacyclic fields, Canad. Math. Bull. 36(1) (1993), pp. 103–107.
[6] H. Reichart, Arithmetische Theorie der kubischen Zahlköper als Radikalkörper, Monatsh. Math. Phys. 40 (1933), pp. 323–350.
[7] A. Scholz und O. Taussky, Die Hauptideale der kubischen Klassenk¨orper imagin¨ar-quadratischer Zahlk¨orper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenk¨orperturm, J. Reine Angew. Math. 171 (1934), pp. 19–41.
[8] S. Selmane, Quadratic extensions of totally real quintic fields, Math. Comp. 70, 234 (2001), pp. 837–843
[9] S. Selmane, Tenth degree number fields containing quintic subfields of signature (3,1), the Acta Mathematica et Informatica Universitatis Ostraviensis.
[10] S. Selmane, Tenth degree number fields with quintic fields having one real place, Math. Comp. 70, 234 (2001), pp. 845–851.
[11] www.megrez.math.u-bordeaux.fr.