理想剩余类群上Gauss定理的推广

Generlization of Gauss' theorem on the groups of residue classes mod m.

利用代数数论的方法,把整数环上的Gauss定理推广到理想m的剩余类群R(m)上,得到同余式,∏bij∈R( m)1≤i1 <i2 <…<ik≤hbi1 bi2 …bik ≡1(m od m) , 若e(2 )≠1,(- 1) Ck- 1 φ( m) - 1 (mod m) , 若e(2 ) =1,其中e(2 )是R(m)中阶为2的幂次的基的个数,h =φ(m)是R(m)中与m互素的理想个数.

By using the method of algebraic number theory, the following congruence is proved=∏b_ i-j ∈R(m)1≤i-1<i-2<:<i-k≤hb_ i-1 b_ i-2 :b_ i-k ≡1(mod m), e(2)≠1,=(-1) C k-1 _ φ(m)-1 (mod m), e(2)=1,=e(2) is the number of those basis elements whose order is the power of 2, h is the number of elements in R(m), h equals Euler's function φ(m), and φ(m)is the number of ideals which are relative prime to m.