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Published June 1977 | public
Journal Article

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


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.

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.

Additional details

August 22, 2023
October 20, 2023