Higher composition laws III : The parametrization of quartic rings
نویسندگان
چکیده
In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized. In particular, a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state. In the quadratic case, one need only note that a quadratic ring—i.e., any ring that is free of rank 2 as a Z-module—is uniquely specified up to isomorphism by its discriminant; and conversely, given any discriminant D, i.e., any integer congruent to 0 or 1 (mod 4), there is a unique quadratic ring having discriminant D, namely
منابع مشابه
Higher composition laws and applications
In 1801 Gauss laid down a remarkable law of composition on integral binary quadratic forms. This discovery, known as Gauss composition, not only had a profound influence on elementary number theory but also laid the foundations for ideal theory and modern algebraic number theory. Even today, Gauss composition remains one of the best ways of understanding ideal class groups of quadratic fields. ...
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