Construction of Generic Polynomials in Galois Theory and application to Number Theory

由于目前的助学金，我们能够组织七次研究研讨会，邀请该领域最活跃的数学家，通过这些研讨会，我们对我们的主题进行了许多讨论。这使我们能够沿着伽罗瓦理论的研究项目取得相当大的进展。对于构造Q上给定有限群的一般多项式的主题，我们的第一个结果是对5次传递置换群中的每一个都构造具有两个参数的具体的、简单的五次多项式族。作为一个显著的应用，我们建立了a . bumer发现的著名的6次多项式族A_5的一般证明，它与雅可比矩阵为5的实乘法的2属代数曲线有关。我们的第二个结果是关于Z/8Z上仿射变换群的子群，幂为8的元阿贝尔群的Noethers问题。我们证明了它们每一个的4次线性表示的肯定答案，而对8次循环群的否定答案。作为这一结果的双积，我们得到了4次循环扩展L/K嵌入到8次循环扩展中的一个简单准则。

Thanks to the current Grant-in-Aid, we were able to organize seven research workshops inviting the most active mathematicians on this field, through which we had many discussions on our subjects.This enabled us to make a considerable developments along our reseach project on Galois theory.As for the main theme of constructing generic polynomials with given finite groups over Q, our first result is the construction of concrete and simple families of quintic polynomials with two parameters for each of the five transitive permutation groups of degree 5. As a remarkable application we have established the proof of the genericity of the famous family of A_5 polynomials of degree 6 found by A.Bumer, in connection with algebraic curves of genus two whose Jacobian have real multiplication of discriminant 5.Our second result is concerned with the Noethers' Problem for the meta abelian groups of exponent 8 which are subgroups of the affine transformation group over Z/8Z. We have proved the affirmative answer for the linear representation of degree 4 for each of them, in contrast with the negative answer for cyclic group of order 8. As a biproduct of this result, we obtained a simple criterion for a cyclic extension L/K of degree 4 to be embedded into a cyclic extension of degree 8.