四次域Q(m^(1/2),n^(1/2))的正规整基的存在性

Existence of the normal integral basis of the quartic field

对Galois数域存在正规整基的充要条件进行研究具有一定的理论价值。四次域是重要的Galois数域之一,该文对四次域Q(m,n)进行了讨论;利用Galois扩张、域的判别式及整基等一些理论,得出了其具有正规整基的充要条件为m≡n≡1(mod4)(其中m,n是2个不同的无平方因子有理整数)。

The quartic field is an important one among Galois number fields. This paper deals with the necessary and sufficient condition for the existence of the normal integral basis of the quartic field. According to the Galois extension theory of the number field, the discriminant of the field and the integral basis theory, the necessary and sufficient condition is attained which is m≡n≡1（mod4）, where rn and n are two different rational integers without square factors.