A Guide to Elementary Number Theory: Gauss's Lemma

Gauss's Lemma is needed to prove the Quadratic Reciprocity Theorem, that for odd primes p and q , (p/q) = (q/p) unless p ≡ q ≡ 3 (mod 4), in which case (p/q) = -(q/p), but it also has other uses. Theorem (Gauss's Lemma) Suppose that p is an odd prime, p ∤ a, and that among the least residues (mod p ) of a, 2a , …, (( p -1)/2)a exactly g are greater than (p - 1)/2. Then (a/p) = (-1) g . Proof Divide the least residues (mod p ) of a, 2a , …, (( p - 1)/2)a into two classes: r 1 , r 2 , …, r k that are less than or equal to ( p - 1)/2 and s 1 , s 2 , …, s g that are greater than ( p - 1)/2. Then k + g = ( p - 1)/2. By Euler's Criterion, to prove the theorem it is enough to show that a ( p -1)/2 ≡ (-1) g (mod p ). No two of r 1 , r 2 , …, r k are congruent (mod p ). If they were we would have k 1 a ≡ k 2 a (mod p ) and, because (a, p) = 1, k 1 ≡ k 2 (mod p ). Because k 1 and k 1 are both in the interval [1, ( p -1)/2] we have k 1 = k 1 .