A Counting Proof for When 2 Is a Quadratic Residue

Abstract

Using the group consisting of the eight Möbius transformations x, – x, $1/x,-1/x$, $(x-1)/(x+1),(x+1)/(1-x)$, $(x+1)/(x-1)$, and $(1-x)/(x+1)$, we present an enumerative proof of the classical result for when the element 2 is a quadratic residue in the finite field F_q .

• MSC: Primary 11A07

Acknowledgmenst

The authors thank David Leep for suggestions that improved the exposition of an earlier version of this note. This work was partially supported by a grant from the Simons Foundation (#429370 to Richard Ehrenborg).