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The Trace Form Over Cyclic Number Fields

Published online by Cambridge University Press:  14 April 2020

Wilmar Bolaños
Affiliation:
Department of Mathematics, Universidad de los Andes, Bogotá, Colombia e-mail: [email protected]
Guillermo Mantilla-Soler*
Affiliation:
Guillermo Mantilla-Soler, Department of Mathematics, Universidad Konrad Lorenz, Bogotá, Colombia, and Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland

Abstract

In the mid 80’s Conner and Perlis showed that for cyclic number fields of prime degree p the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.

Type
Article
Copyright
© Canadian Mathematical Society 2020

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References

Bayer-Fluckiger,, E. Galois Cohomology and the Trace form. Jahresber. Deutsch. Math.-Verein 962(1994), 3555.Google Scholar
Bhargava, M., Shankar, A. Wang, and X Squarefree values of polynomial discriminants I. Preprint, 2016. arXiv:1611.09806.Google Scholar
Bhargava and, M. Shnidman,, A. On the number of cubic orders of bounded discriminant having automorphism group $C3$ and related problems. Algebra. Number Theory, 81(2014), 5388.CrossRefGoogle Scholar
Cassels,, J. Rational quadratic forms . Dover Publications, Inc., Mineola, NY 2008.Google Scholar
Conner, P. and Perlis,, R. A survey of trace forms of algebraic number fields. World Scientific, Singapore, 1984.CrossRefGoogle Scholar
Ellenberg, J. S. and Venkatesh,, A. The number of extensions of a number field with fixed degree and bounded discriminant. Ann. of Math. 163(2), 723741 (2006).CrossRefGoogle Scholar
A. Frohlich and M. Taylor, Algebraic number theory. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1991.Google Scholar
Harron, P. and Harron, R., The shapes of Galois quartic fields. Preprint, 2019. arXiv:1908.03969.CrossRefGoogle Scholar
Mantilla-Soler, G, On the arithmetic determination of the trace. J. Algebra 444(2015), 272283.CrossRefGoogle Scholar
Mantilla-Soler, G. and Rivera-Guaca,, C. An introduction to Casimir pairings and some arithmetic applications. Preprint, 2019. arXiv:1812.03133v3.Google Scholar
Narkiewicz,, W. Elementary and analytic theory of algebraic numbers . 3rd edition, Springer-Verlag, Berlin Heidelberg, New York, 2004.CrossRefGoogle Scholar
Neukirch,, J. Algebraische Zahlentheorie. Springer-Verlag, Berlin Heidelberg, 2007.Google Scholar
Schmidt, W. M, Number fields of given degree and bounded discriminant. Astérisque 228 (1995), 189195. Columbia University Number Theory Seminar (New York, 1992).Google Scholar
Serre,, J. Local fields . Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979.Google Scholar
Serre,, J. L’invariant de Witt de la forme $\textit{Tr}\left({x}^2\right)$ . Comm. Math. Helv. 27 (1984), 651.CrossRefGoogle Scholar
Taussky, O., The discriminant matrix of a number field . J. London. Math. Soc. 43 (1968), 152154.CrossRefGoogle Scholar
Wright,, D. J. Distribution of discriminants of abelian extensions. Proceedings of the London Mathematical Society, 3(1989), 1750.CrossRefGoogle Scholar