A Characterization of One-Element $p$-Bases of Rings of Constants
Volume 59 / 2011
Bulletin Polish Acad. Sci. Math. 59 (2011), 19-26
MSC: Primary 13N15; Secondary 12E05.
DOI: 10.4064/ba59-1-3
Abstract
Let $K$ be a unique factorization domain of characteristic $p>0$, and let $f\in K[x_1,\dots,x_n]$ be a polynomial not lying in $K[x_1^p,\dots,x_n^p]$. We prove that $K[x_1^p,\dots,x_n^p, f]$ is the ring of constants of a $K$-derivation of $K[x_1,\dots,x_n]$ if and only if all the partial derivatives of $f$ are relatively prime. The proof is based on a generalization of Freudenburg's lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.
