# Sum of n-th roots is rarely rational

Let $m,n$ be positive integers, and $\displaystyle \Phi_{m,n}~:~ {\mathbb{R}_+^*}^m \to \mathbb{R}_+^*, \ \ \ (x_1,x_2, \ldots , x_m) \mapsto \sum_{k=1}^m \sqrt[n]{x_k}$.

Clearly for $m=1$ if for all positive integer $n$, we have $\Phi_{1,n}(x) \in \mathbb Q$, then $x=1$.

It seems that the same conclusion holds for $m>1$ (or at least the subset of ${\mathbb{R}_+^*}^m$ for which $\Phi_{m,n}(x) \in \mathbb Q$ is finite).

Is it true (or even obvious and I missed it)?

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How does your definition of Phi depend on n? – Kevin Ventullo May 8 '10 at 21:16
@Kevin: Those are n-th roots. – JBL May 8 '10 at 21:58

The following conclusion is true: If $\Phi_{m,n}(x)\in\mathbb{Q}$ for all positive integers n, then x1=x2=...=xn=1.