# On a problem of Perron

## TL;DR

This paper characterizes the set of squares in finite fields using additive properties, providing a new purely additive perspective on Perron's problem.

## Contribution

It offers a novel additive characterization of squares in finite fields, linking set partitions to algebraic properties.

## Key findings

- Partition into squares and non-squares is characterized by additive properties.
- Provides a purely additive criterion for identifying squares in finite fields.
- Connects set partitions with algebraic structure in finite fields.

## Abstract

It is shown that a partition $\mathfrak A\cup \mathfrak B$ of the set $\mathbb F_{p^m}^*=\mathbb F_{p^m}-\{0 \}$, 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 $\mathbb F_{p^m}$.