tailieunhanh - Đề tài " Higher composition laws IV: The parametrization of quintic rings "

In the first three parts of this series, we considered quadratic, cubic and quartic rings (., rings free of ranks 2, 3, and 4 over Z) respectively, and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space. These orbit results are summarized in Table 1. In particular, the theories behind the parametrizations of quadratic, cubic, and quartic rings, noted in items #2, 9, and 13 of Table 1, were seen to closely parallel the classical developments of the solutions to the quadratic,. | Annals of Mathematics Higher composition laws IV The parametrization of quintic rings By Manjul Bhargava Annals of Mathematics 167 2008 53 94 Higher composition laws IV The parametrization of quintic rings By Manjul Bhargava 1. Introduction In the first three parts of this series we considered quadratic cubic and quartic rings . rings free of ranks 2 3 and 4 over Z respectively and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space. These orbit results are summarized in Table 1. In particular the theories behind the parametrizations of quadratic cubic and quartic rings noted in items 2 9 and 13 of Table 1 were seen to closely parallel the classical developments of the solutions to the quadratic cubic and quartic equations respectively. Despite the quintic having been shown to be unsolvable nearly two centuries ago by Abel it turns out there still remains much to be said regarding the integral theory of the quintic. Although a solution naturally still is not possible we show in this article that it is nevertheless possible to completely parametrize quintic rings indeed a theory just as complete as in the quadratic cubic and quartic cases exists also in the case of the quintic. In fact we present here a unified theory of ring parametrizations which includes the cases n 2 3 4 and 5 simultaneously. Our strategy to parametrize rings of rank n is as follows. To any order R in a number field of degree n we give a method of attaching to R a set of n points XR c Pn-2 C which is well-defined up to transformations in GLn-i Z . We then seek to understand the hypersurfaces in Pra 2 C defined over Z and of smallest possible degree which vanish on all n points of XR. We find that the hypersurfaces over Z passing through all n points in XR correspond in a remarkable way to functions between R and certain resolvent rings a notion we introduced in 1 and 4 . .