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

由于这次的补助金，我们得以组织了七个研究工作坊，邀请了这一领域最活跃的数学家，通过这些工作坊，我们就我们的课题进行了许多讨论。这使我们能够沿着我们的伽罗瓦理论研究项目取得相当大的进展。至于用给定的有限群构造一般多项式的主题，我们的第一个结果是，对于五个5次可迁置换群中的每一个，构造了具有两个参数的五次多项式的具体而简单的族。作为一个显著的应用，我们建立了由A.Bumer发现的著名的六次A_5多项式族的通有性的证明，它与其Jacobian具有判别式5的真实的乘法的亏格为2的代数曲线有关。我们的第二个结果涉及Z/8 Z上仿射变换群的子群--指数为8的Meta阿贝尔群的Noethers问题。我们已经证明了4阶线性表示的肯定答案，与之相反，8阶循环群的否定答案。作为这个结果的一个双积，我们得到了一个简单的准则，一个循环扩张L/K的程度4嵌入到一个循环扩张的程度8。

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.