Prime Numbers

2.3 Squares and roots 99
Definition 2.3.6. The quadratic Gauss sum G(a; m) is defined for integers
a, N, withN positive, as
G(a; N) =
N−1
j=0
e 2πiaj2 /N .
This sum is—up to conjugation perhaps—a discrete Fourier transform (DFT)
as used in various guises in Chapter 8.8. A more general form—a character
sum—is used in primality proving (Section 4.4). The central result we wish
to cite makes an important connection with the Legendre symbol:
Theorem 2.3.7 (Gauss). For odd prime p and integer a ≡ 0(modp),
a
G(a; p) = G(1; p),
p
and generally, for positive integer m,
G(1; m) = 1√
m
m(1 + i)(1 + (−i) ).
2
The first assertion is really very easy, the reader might consider proving it
without looking up references. The two assertions of the theorem together
allow for Fourier inversion of the sum, so that one can actually express the
Legendre symbol for a ≡ 0(modp) by
a
=
p
c p−1
√ e
p
2πiaj2 /p c
= √p
j=0
p−1
j=0
j
e
p
2πiaj/p , (2.12)
where c = 1, −i as p ≡ 1, 3 (mod 4), respectively. This shows that the
Legendre symbol is, essentially, its own discrete Fourier transform (DFT).
For practice in manipulating Gauss sums, see Exercises 1.66, 2.27, 2.28, and
9.41.
2.3.2 Square roots
Armed now with algorithms for gcd, inverse (actually the −1 power), and
positive integer powers, we turn to the issue of square roots modulo a prime.
As we shall see, the technique actually calls for raising residues to high integral
powers, and so the task is not at all like taking square roots in the real
numbers.
We have seen that for odd prime p, the solvability of a congruence
x 2 ≡ a ≡ 0(modp)
is signified by the value of the Legendre symbol
a
a
p . When p = 1, an
important problem is to find a “square root” x, of which there will be two,
one the other’s negative (mod p). We shall give two algorithms for extracting