论文标题
在几个变量中的理性函数领域的根本扩展上
On a radical extension of the field of rational functions in several variables
论文作者
论文摘要
令$ f $为一个字段，让$ f（x_1，\ dots，x_n）$为$ n $变量中的有理函数字段$ x_1，\ dots，x_n $ over $ f $。令$ t = x_1+\ cdots+x_n \在f（x_1，\ dots，x_n）$中，让$ m $为正整数，使得$ \ text {char} \，f \ nmid m $。是否可以在$ x_1^m \ dots，x_n^m $和$ f $上的$ x_1^m \ dots中表达每个$ x_i $？证明这可以做到并不难，但是展示如何完成这是另一回事。我们使用非构造证明和建设性证据肯定地回答了上述问题。
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.
