# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Purely inseparable extensions of $\textbf {k}[X,Y]$HTML articles powered by AMS MathViewer

by D. Daigle
Proc. Amer. Math. Soc. 121 (1994), 1-12 Request permission

## Abstract:

Let k be a field of characteristic $p > 0$ and R a polynomial ring in two variables over k. Define weak variable of R to mean an element u of R such that $u - \lambda$ is irreducible for each $\lambda \in {\mathbf {k}}$ and such that ${R^{{p^n}}} \subseteq {\mathbf {k}}[u,v]$ for some $v \in R$ and some integer $n \geq 0$. Given a weak variable u of R, consider all $v \in R$ such that ${R^{{p^n}}} \subseteq {\mathbf {k}}[u,v]$ for some n; if one of these v is "absolutely smaller" than u (roughly, ${\deg _X}v < {\deg _X}u$ for all coordinate systems (X, Y) of R), we call it an R-companion of u. The main result gives a connection between the structure of a purely inseparable extension $R \supset A$, where A is a polynomial ring in two variables, and whether or not there exists a companion for each u in a suitable set of weak variables of R.
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