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. doi:10.1007/bf01241822. https://resolver.caltech.edu/CaltechAUTHORS:20201013-104023016
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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 | |||||||||
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| 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. | |||||||||
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| Subject Keywords: | Class Group; Ideal Class; Quadratic Field; Ideal Class Group | |||||||||
| Issue or Number: | 2 | |||||||||
| DOI: | 10.1007/bf01241822 | |||||||||
| 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: | 16 Nov 2021 18:49 |
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