Number Theory: An Introduction to Mathematics

136 III More on Divisibility

or, puttingv=uw,

τ^2 =

∑

w≡0modp

(w/p)

∑

u≡0modp

ζu(^1 +w).

Since the coefficients oftp−^1 andtp−^2 inf(t)are 1, the sum of the roots is−1 and thus
∑

u≡0modp

ζau=−1ifa≡0modp.

On the other hand, ifa≡0modp,thenζau=1and
∑

u≡0modp

ζau=p− 1.

Hence

τ^2 =(− 1 /p)(p− 1 )−

∑

w≡ 0 ,−1modp

(w/p)=(− 1 /p)p−

∑

w≡0modp

(w/p).

Since there are equally many quadratic residues and quadratic nonresidues, the last
sum vanishes and we obtain finally

τ^2 =(− 1 )(p−^1 )/^2 p.

Thusτ=0 and from the previous expression forτqwe now obtain

τq−^1 =(q/p).

But

τq−^1 =(τ^2 )(q−^1 )/^2 ={(− 1 )(p−^1 )/^2 p}(q−^1 )/^2

andp(q−^1 )/^2 =(p/q), by Proposition II.28 again. Hence

(q/p)=(− 1 )(p−^1 )(q−^1 )/^4 (p/q),

which is the law of quadratic reciprocity.
The preceding proof is a variant of the sixth proof of Gauss (1818). Already in
1801 Gauss had shown that ifpis an odd prime, then

∑p−^1

k= 0

e^2 πik

(^2) /p
=±

√

por±i

√

paccording asp≡1orp≡3 mod 4.

After four more years of labour he managed to show that in fact the+signs must be
taken. From this result he obtained his fourth proof of the law of quadratic reciprocity.
The sixth proof avoided this sign determination, but Gauss’s result is of interest in it-
self. Dirichlet (1835) derived it by a powerful analytic method, which is readily gener-
alized. Although we will make no later use of it, we now present Dirichlet’s argument.
For any positive integersm,n,wedefinetheGauss sum G(m,n)by