CaltechAUTHORS
  A Caltech Library Service

Norms from quadratic fields and their relation to noncommuting 2×2 matrices III. A link between the 4-rank of the ideal class groups in ℚ(√m) and in ℚ(√-m)

Taussky, Olga (1977) Norms from quadratic fields and their relation to noncommuting 2×2 matrices III. A link between the 4-rank of the ideal class groups in ℚ(√m) and in ℚ(√-m). Mathematische Zeitschrift, 154 (2). pp. 91-95. ISSN 0025-5874. https://resolver.caltech.edu/CaltechAUTHORS:20201013-104023016

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20201013-104023016

Abstract

This paper is concerned with the representation of an integral 2 x 2 matrix A as A = S₁S₂ with S_i = S'_i and integral and facts connected with it. In [6] the following was shown. If the characteristic polynomial of A is x²-m with m square free and ≡2 or 3(4) then a factorization of A as above is only possible if the ideal class in Z[√m[ associated with A is of order a factor of 4. If the ideal class is of order 4 then the S_i cannot be unimodular. Now it is shown that a factorization for an A with characteristic polynomial x²-m, m square free, leads to an ideal class in the narrow sense of order 4 in Z[√-m]. This is achieved by associating with the factorization an integral ternary form representing zero in a nontrivial way. The conditions for this to happen are known.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/bf01241822DOIArticle
https://rdcu.be/b8sJpPublisherFree ReadCube access
Additional Information:© 1977 Springer-Verlag. To Günter Pickert, on his sixtieth birthday. Received September 28, 1976. The author is indebted to J.W.S. Cassels, A. Fröhlich, H. Kisilevsky for helpful discussions. This work was carried out (in part) under an NSF contract.
Funders:
Funding AgencyGrant Number
NSFUNSPECIFIED
Subject Keywords:Class Group; Ideal Class; Quadratic Field; Ideal Class Group
Issue or Number:2
Record Number:CaltechAUTHORS:20201013-104023016
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20201013-104023016
Official Citation:Taussky, O. Norms from quadratic fields and their relation to noncommuting 2×2 matrices III. A link between the 4-rank of the ideal class groups in Q(√m) and in Q (√−m) . Math Z 154, 91–95 (1977). https://doi.org/10.1007/BF01241822
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106017
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:13 Oct 2020 20:25
Last Modified:13 Oct 2020 20:28

Repository Staff Only: item control page