Cogredient Standard Forms of Symmetric Matrices over Galois Rings of Odd Characteristic

Let R = GR ( p s , p s m) be a Galois ring of characteristicp sand cardinalityp s m, where s and m are positive integers and p is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over R are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over R is obtained.