局部域有限阿贝尔扩张的导子与其判别式的应用

Conductor and Discriminant of Finite Abelian Extension of Local Field and Its Application

在文献[1]的基础上,根据有限阿贝尔扩张的范数群及上限数分歧群的理论,进一步研究阿贝尔扩张k′/k的导子f(χ)与其判别式D(k′/k)之间的关系,从而得到局部域的有限阿贝尔扩张的又一个应用,即有限阿贝尔扩张k′/k的判别式D(k′/k)=∏χf(χ),其中上式右端是取遍Gal(k′/k)的所有特征标χ的导子之积.

The finite abelian extension of local fields was applied to the finite ramification groups and the norm residue map of local field and the canonical homomorphism.The results of the application are Gr=ρk′/k（Ui） and Gr=Gi,where Gr is the ramification groups of the local finite extension in the upper numbering;i is the integer satisfying,r-1r≤i,i≥0 when r≥-1.On the foundation of above applications,the paper further studies the relations between the conductor f（χ） of abelian extension k′/k,and its discriminant,according to the theories of the norm groups on the finite abelian extension and the ramification groups in the upper numbering.Therefore we obtain the another application in the finite abelian extension of local field,that is,D（k′/k） is the discriminant of the finite abelian extension k′/k.Then D（k′/k）=∏χf（χ）,where the product is taken over all characters χ of Gal（k′/k）.