On the structure of the Group-ring of an elementary abelian group and a Theorem of Brauer and Nesbitt

Abstract

Let G be an elementary abelian p-group generated by m elements and let F be a finite field of characteristic p. We study the structure of the group-ring FG. It is observed that FG is isomorphic to a certain quotient ring of a polynomial ring over F in m indeterminates. In this isomorphic copy, a chain of ideals is defined such that each ideal is left invariant, under the natural action of the group GL(m,p). We study the structure of these ideals and prove some results. The quotient of any two successive elements in this chain of ideals is isomorphic to the ring of homogeneous polynomials in m variables over F of a certain degree. This observation leads us to give an elementary proof of a Theorem of Brauer and Nesbitt (Annals of Math (2) 42 (1941), 556-590) that the ring of homogeneous polynomials over F in m variables of a given degree is irreducible under the natural action of the group GL(m,p).