A Characterization of One-Element $p$-Bases of Rings of Constants

Piotr Jędrzejewicz¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University 87-100 Toruń, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) no. 1, pp. 19-26.

Résumé

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.

Détail

DOI : 10.4064/ba59-1-3

Keywords: unique factorization domain characteristic dots polynomial lying dots prove dots ring constants k derivation dots only partial derivatives relatively prime proof based generalization freudenburgs lemma polynomials unique factorization domain arbitrary characteristic