Prime Numbers

40 Chapter 1 PRIMES!
Definition 1.4.4. Suppose D is a positive integer and χ is a function from
the integers to the complex numbers such that
(1) For all integers m, n, χ(mn) =χ(m)χ(n).
(2) χ is periodic modulo D.
(3) χ(n) = 0 if and only if gcd(n, D) > 1.
Then χ is said to be a Dirichlet character to the modulus D.
For example, if D>1 is an odd integer, then the Jacobi symbol
n
D is a
Dirichlet character to the modulus D (see Definition 2.3.3).
It is a simple consequence of the definition that if χ is a Dirichlet character
(mod D) and if gcd(n, D) = 1, then χ(n) ϕ(D) =1;thatis,χ(n) is a root of
unity. Indeed, χ(n) ϕ(D) = χ nϕ(D) = χ(1), where the last equality follows
from the Euler theorem (see (2.2)) that for gcd(n, D) =1wehavenϕ(D) ≡ 1
(mod D). But χ(1) = 1, since χ(1) = χ(1) 2 and χ(1) = 0.
If χ1 is a Dirichlet character to the modulus D1 and χ2 is one
to the modulus D2, then χ1χ2 is a Dirichlet character to the modulus
lcm [D1,D2], where by (χ1χ2)(n) we simply mean χ1(n)χ2(n). Thus, the
Dirichlet characters to the modulus D are closed under multiplication. In
fact, they form a multiplicative group, where the identity is χ0, the “principal
character” to the modulus D. Wehaveχ0(n) =1whengcd(n, D) = 1, and 0
otherwise. The multiplicative inverse of a character χ to the modulus D is its
complex conjugate, χ.
As with integers, characters can be uniquely factored. If D has the prime
factorization p a1
1 ···pak k , then a character χ (mod D) can be uniquely factored
as χ1 ···χk, whereχjis a character (mod p aj
j ).
In addition, characters modulo prime powers are easy to construct and
understand. Let q = pa be an odd prime power or 2 or 4. There are primitive
roots (mod q), say one of them is g. (A primitive root for a modulus D is a
cyclic generator of the multiplicative group Z∗ D of residues modulo D that are
coprime to D. This group is cyclic if and only if D is not properly divisible
by 4 and not divisible by two different odd primes.) Then the powers of g
(mod q) run over all the residue classes (mod q) coprime to q. So,ifwepick
a ϕ(q)-th root of 1, call it η, then we have picked the unique character χ
(mod q) withχ(g) =η. We see there are ϕ(q) different characters χ (mod q).
It is a touch more difficult in the case that q =2a with a>2, since
then there is no primitive root. However, the order of 3 (mod 2a )fora>2is
always 2a−2 ,and2a−1 + 1, which has order 2, is not in the cyclic subgroup
generated by 3. Thus these two residues, 3 and 2a−1 + 1, freely generate the
multiplicative group of odd residues (mod 2a ). We can then construct the
characters (mod 2a )bychoosinga2a−2-throotof1,sayη, and choosing
ε ∈{1, −1}, and then we have picked the unique character χ (mod 2a )with
χ(3) = η, χ(2a−1 +1)=ε. Again there are ϕ(q) characters χ (mod q).
Thus, there are exactly ϕ(D) characters (mod D), and the above proof
not only lets us construct them, but it shows that the group of characters