On a problem of Perron

Abstract

It is shown that a partition $\mathfrak{A}\cup \mathfrak{B}$ of the set ${\mathbb{F}}_{p^m}^{\ast }={\mathbb{F}}_{p^m}-\{0\}$, with $|\mathfrak{A}|=|\mathfrak{B}|$, is the separation into squares and non squares, if and only if the elements of $\mathfrak{A}$ and $\mathfrak{B}$ satisfy certain additive properties, thus providing a purely additive characterization of the set of squares in ${\mathbb{F}}_{p^m}$.

7 pages