## Purely inseparable extensions of $\textbf {k}[X,Y]$

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**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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## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**121**(1994), 1-12 - MSC: Primary 13F20; Secondary 13B02
- DOI: https://doi.org/10.1090/S0002-9939-1994-1227516-X
- MathSciNet review: 1227516