Number fields with solvable Galois groups and small Galois root discriminants

Jones, John; Wallington, Rachel

doi:10.1090/S0025-5718-2011-02511-1

Number fields with solvable Galois groups and small Galois root discriminants
HTML articles powered by AMS MathViewer

by John W. Jones and Rachel Wallington PDF

Math. Comp. 81 (2012), 555-567 Request permission

Abstract:

We apply class field theory to compute complete tables of number fields with Galois root discriminant less than $8\pi e^{\gamma }$. This includes all solvable Galois groups which appear in degree less than $10$, groups of order less than $24$, and all dihedral groups $D_p$ where $p$ is prime.

References

Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
Claus Fieker and Jürgen Klüners, Minimal discriminants for fields with small Frobenius groups as Galois groups, J. Number Theory 99 (2003), no. 2, 318–337. MR 1968456, DOI 10.1016/S0022-314X(02)00071-9
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4, 2006, \verb+(http://www.gap-system.org)+.
John W. Jones, Wild ramification bounds and simple group Galois extensions ramified only at 2, Proc. Amer. Math. Soc. 139 (2011), no. 3, 807–821. MR 2745634, DOI 10.1090/S0002-9939-2010-10628-7
John W. Jones and David P. Roberts, Septic fields with discriminant $\pm 2^a3^b$, Math. Comp. 72 (2003), no. 244, 1975–1985. MR 1986816, DOI 10.1090/S0025-5718-03-01510-2
John W. Jones and David P. Roberts, Galois number fields with small root discriminant, J. Number Theory 122 (2007), no. 2, 379–407. MR 2292261, DOI 10.1016/j.jnt.2006.05.001
—, Website: Number fields with small grd, http://hobbes.la.asu.edu/lowgrd, 2007.
The PARI Group, Bordeaux, Pari/gp, version 2.3.4, 2008.
Robert Perlis, On the equation $\zeta _{K}(s)=\zeta _{K’}(s)$, J. Number Theory 9 (1977), no. 3, 342–360. MR 447188, DOI 10.1016/0022-314X(77)90070-1
Leila Schneps, $\~D_4$ et $\hat D_4$ comme groupes de Galois, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 2, 33–36 (French, with English summary). MR 980080
Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
Jean-Pierre Serre, Œuvres. Vol. III, Springer-Verlag, Berlin, 1986 (French). 1972–1984. MR 926691