A Description Of Galois Group And On Some Matrix Equations

In the first part of this paper, we give a matrix representation of a Galois group. It is well known that an extension field is a linear space over basic field, when the extension field is a Galois extension, every Galois action can be seen a linear transformation. By finding a set of basis, we use matrices of transformation under a set of basis to describe the structure of the Galois group. If a normal extension has a normal basis, then it is simpler to describe the structure of Galois group in this method. Moreover, we give the relation of two bases.In the second part, we study the solution of some matrix equations in field theory and module theory,i.e., it is determined whether matrix equations have solutions in a field. For different polynomials g(x), if the characteristic polynomial of a n matrix A is irreducible, then we get some theorems to determine matrix equations g(x) =A solvable; if it is reducible, then, to see n-dimension space vectors M over a field F as F[x]-module, we use module theory to determine these equations solvable such that it is simpler and clearer to investigate these questions.