有限局部环Z/p^mZ上3阶交错矩阵的计数定理

An Anzahl Theorem of 3×3 Matrices over Finite Local Ring

设H是有限局部环Z/pmZ上的3×3交错矩阵,通过确定H的标准形,计算出有限局部环Z/pmZ上合同标准形的3×3交错矩阵的个数nk,其中当0≤k<m时,nk=p3m-p3(m-1)/p3k;当k=m时,nk=1。

Let H be a 3 × 3 alternate matrices over finite local ring Z/p^mZ. By determining the normal form of H,nk is computed,and nk is the number of 3 × 3 alternate matrices being cogerdient to the normal , form,where 0≤k〈m时,nk=[p^3m-p^3（m-1）]/p^3k,if 0≤k〈m;nk=1,if k=m.