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

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.