Field degrees and multiplicities for non-integral extensions

Abstract

Let $R$ be a graded subalgebra of a polynomial ring $S$ over a field so that $S$ is algebraic over $R$. The goal of this paper is to relate the generator degrees of $R$ to the degree $[S:R]$ of the underlying quotient field extension, and to provide a numerical criterion for $S$ to be integral over $R$ that is based on this relationship. As an application we obtain a condition guaranteeing that a ring of invariants of a finite group is a polynomial ring.