On a radical extension of the field of rational functions in several variables

TL;DR

This paper investigates whether the variables in a multivariate rational function field can be expressed using the sum of variables and their m-th powers, providing both nonconstructive and constructive proofs for this extension.

Contribution

It introduces a novel extension of the field of rational functions by expressing variables in terms of their m-th powers and the sum, with explicit constructive methods.

Findings

Variables can be expressed as rational functions of the sum and m-th powers.

The paper provides both nonconstructive and constructive proofs.

The extension is possible under the condition that the characteristic of F does not divide m.

Abstract

Let $F$ be a field and let $F(X_1,\dots,X_n)$ be the field of rational functions in $n$ variables $X_1,\dots,X_n$ over $F$. Let $T=X_1+\cdots+X_n\in F(X_1,\dots,X_n)$ and let $m$ be a positive integer such that $\text{char}\,F\nmid m$. Is it possible to express each $X_i$ as a rational function in $X_1^m\dots,X_n^m$ and $T$ over $F$? It is not difficult to prove that this can be done but it is another matter to show how this is done. We answer the above question affirmatively with a nonconstructive proof and a constructive proof.