An Exploration of Quadratic Residues in Finite Fields

Abstract

If an element in a given field can be expressed as a product of two equivalent elements that are also in the field, then this element is a quadratic residue of that field. For example, the set of quadratic residues of the field of rational numbers are the ratios of perfects squares. In the real numbers, the quadratic residues are the nonnegative numbers, and in the complex field, every number is a quadratic residue. We will explore quadratic residues of finite fields, which are fields with a finite number of elements. Finite fields operate under modular arithmetic. Furthermore, we can construct any finite field using factor rings of polynomial rings. We will explore relationships between finite fields and their quadratic residues, including similarities and differences between finite and the more familiar infinite fields.