Abstract
For each permutation group G on n letters with n ≤ 4, we give results, conjectures and numerical computations on discriminants of number fields L of degree n over ℚ such that the Galois group of the Galois closure of L is isomorphic to G.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Belabas, K.: On the mean 3-rank of quadratic fields. Compositio Math. 118, 1–9 (1999)
Belabas, K.: A fast algorithm to compute cubic fields. Math. Comp. 66, 1213–1237 (1997)
Cohen, H., Diaz y Diaz, F., Olivier, M.: Density of number field discriminants (in preparation)
Cohen, H., Diaz y Diaz, F., Olivier, M.: Construction of tables of quartic fields using Kummer theory. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 257–268. Springer, Heidelberg (2000)
Cohen, H.: A course in computational algebraic number theory (third printing), GTM, vol. 138. Springer, Heidelberg (1996)
Cohen, H.: Advanced topics in computational number theory, GTM, vol. 193. Springer, Heidelberg (2000)
Datskovsky, B., Wright, D.J.: Wright, Density of discriminants of cubic extensions. J. reine angew. Math. 386, 116–138 (1988)
Davenport, H., Heilbronn, H.: On the density of discriminants of cubic fields I. Bull. London Math. Soc. 1, 345–348 (1969)
Davenport, H., Heilbronn, H.: On the density of discriminants of cubic fields II. Proc. Royal Soc. A 322, 405–420 (1971)
Roberts, D.: Density of cubic field discriminants, Algebraic number theory preprint archive 177 (April 26, 1999)
Shintani, T.: On Dirichlet series whose coefficients are class numbers of integral binary cubic forms. J. Math. Soc. Japan 24, 132–188 (1972)
Shintani, T.: On zeta-functions associated with the vector space of quadratic forms. J. Fac. Sci. Univ. Tokyo, Sec. 1a 22, 25–66 (1975)
Tenenbaum, G.: Introduction à la théorie analytique et probabiliste des nombres, Cours Spécialisés SMF. Société Mathématique de France 1 (1995)
Wright, D.J.: Distribution of discriminants of Abelian extensions, Proc. London Math. Proc. London Math. Soc. 3(58), 17–50 (1989)
Wright, D.J., Yukie, A.: Prehomogeneous vector spaces and field extensions. Invent. Math. 110, 283–314 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cohen, H., Diaz y Diaz, F., Olivier, M. (2000). Counting Discriminants of Number Fields of Degree up to Four. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_15
Download citation
DOI: https://doi.org/10.1007/10722028_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67695-9
Online ISBN: 978-3-540-44994-2
eBook Packages: Springer Book Archive