Matrix Congruence Over Galois Rings And The Self-dual Bases

Galois rings are a special class of local rings,and all zero divisors with 0 in the ring constitute its only maximal ideal(p),where p is a prime number.This thesis mainly discusses the congruence of symmetric matrices over Galois ring R=GR(ps,m)and the self-dual bases on Galois ring extension R’/R,where R’=GR(ps,mn).When s=1,R’ and R are finite fields Fpmn and Fpm respectively.At present,the congruence of symmetric matrices and the self-dual bases problem over finite fields have been well solved.In this thesis,the following two researches are carried out on the general Galois ring for which s is any positive integer:one is the congruence between symmetric matrices and diagonal matrices over R;another is the sufficient and necessary condition for the existence of self-dual bases and the number of self-dual bases on R’/R when p is odd.Regarding the congruence of symmetric matrices,we divide it into two cases where p is odd and p=2.When p is odd,we show that v as the non-square element in R*can be expressed as the sum of squares of two elements in R*,and obtain the congruence between invertible symmetric matrices and diagonal matrices over R,and give the corresponding examples.Moreover,we give and prove the congruence between general symmetric matrices and diagonal matrices over R.When p=2,we show that if there is an unit in the diagonal elements of invertible symmetric matrix,the matrix is congruent to a special diagonal matrix.Regarding the self-dual bases,firstly,we give a sufficient and necessary condition for the existence of self-dual bases on R’/R when p is odd.In the proof,we mainly use the relevant conclusions for the congruence of matrices obtained in Chapter 3 and the properties of the generalized Frobenius automorphism.Secondly,when p is odd,we give a recursive formula for the number of self-dual bases on R’/R by studying the number of t ×t symmetric matrices of rank r over R.