On the distribution of squares in a finite field

✍ Scribed by Tom Ralston

• Book ID: 104653310
• Publisher: Springer
• Year: 1979
• Tongue: English
• Weight: 215 KB
• Volume: 8
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00181488

✦ Synopsis

Given the Galois field GF (n) where n is odd we may partition the pairs (g,g + 1) where g,g + 1 EGF(n), g(g + 1) ~ 0 into four classes as follows: RR denotes the class of such pairs for which both g and g + 1 are squares in GF(n), RN the pairs for which g is a square and g + 1 a nonsquare, NR the pairs for which g is a nonsquare and g + 1 a square, and NN the pairs for which both g and g + 1 are nonsquares. Raber [6] has recently applied geometric arguments to show that each class contains [(n -I)/4] + E elements, where I'1 ~< 1. The exact size of these classes is also known (see, for example, L.E. Dickson's book, Linear Groups, Dover, 1958, p. 48). In this paper we use arguments similar to those used by Raber to obtain the exact size of each class.