§ XII

Abstract­

Let p be an odd prime; put p=2n+l. Write G for the multiplicative group F _p ^x of congruence classes prime to p modulo p; this has a subgroup H of order 2 consisting of the classes (±1 mod p); we apply to G and H the definitions and lemma of § VIII. If x is an element of G, it belongs to one and only one coset xH; this consists of the two elements (±x mod p); there are n such cosets, viz., the cosets (±1 mod p), (±2 mod p),...(±n mod p). If, in each coset, we choose one element, and if we write these elements, in any order, as u₁,...,u _n , this is known as “a set of representatives” for the cosets of H in G; then every integer prime to p is congruent to one and only one of the integers ±u_1,... ±u _n modulo p. For the purposes of the next lemma, which is due to Gauss and known as Gauss’ lemma, such a set {u₁,...u _n } will be called a “Gaussian set” modulo p. The simplest such set is {1,2,...,n}.