Rings of Constants of the Form k[f]

Abstract

Let k[X] be the algebra of polynomials in n variables over a field k of characteristic zero, and let f ∊ k[X]∖ k. We present a construction of a derivation d of k[X] whose ring of constants is equal to the integral closure of k[f] in k[X]. A similar construction for fields of rational functions is also given.

Key Words:

• Dedekind domains
• Derivations
• Integral closures
• Rings of constants

Mathematics Subject Classification 2000:

• 13N15
• 12H05

ACKNOWLEDGMENTS

The authors wish to thank the referee for his helpful remarks. van den Essen wants to thank the Nicholas Copernicus University for its great hospitality during his stay in February 2001, when this work was initiated. Moulin Ollagnier is very grateful to the same university for the hospitality and the excellent working conditions of the stays during which these topics were mainly discussed.

Notes

Communicated by E. Puczylowski.