## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 51, Number 1 (2007), 299-311.

### Field degrees and multiplicities for non-integral extensions

Bernd Ulrich and Clarence W. Wilkerson

#### 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.

#### Article information

**Source**

Illinois J. Math., Volume 51, Number 1 (2007), 299-311.

**Dates**

First available in Project Euclid: 20 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258735337

**Digital Object Identifier**

doi:10.1215/ijm/1258735337

**Mathematical Reviews number (MathSciNet)**

MR2346199

**Zentralblatt MATH identifier**

1141.13007

**Subjects**

Primary: 13B21: Integral dependence; going up, going down

Secondary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24]

#### Citation

Ulrich, Bernd; Wilkerson, Clarence W. Field degrees and multiplicities for non-integral extensions. Illinois J. Math. 51 (2007), no. 1, 299--311. doi:10.1215/ijm/1258735337. https://projecteuclid.org/euclid.ijm/1258735337