nonprincipal real characters @\symoperatorsmod\tmspace+.1667em\tmspace+.1667emp are unique

Theorem 1

Let p be a prime. Then there is a unique nonprincipal real Dirichlet characterDlmfMathworldPlanetmath χmodp, given by


Proof. Note first that χ(n)=(np) is obviously a nonprincipal real characterPlanetmathPlanetmath modp. Now, suppose χ is any nonprincipal real character modp. Choose some generatorPlanetmathPlanetmathPlanetmath, a, of (/p)*. Clearly χ(a)=-1 (since otherwise χ is principal), and thus χ(ak)=(-1)k. Since ap-1=1, and no lower power of a is 1, it follows that χ is -1 on exactly (p-1)/2 elements of (/p)* and is 1 on exactly (p-1)/2 elements. However, since χ(x2)=χ(x)2=1, χ is 1 on each of the (p-1)/2 squares in (/p)*. Thus χ is 1 on squares and -1 on nonsquares, so χ(n)=(np).

Note that this result is not true if p is not prime. For example, the following is a table of Dirichlet characters modulo 8, all of which are real:

χ1 χ2 χ3 χ4
1 1 1 1 1
3 1 1 -1 -1
5 1 -1 1 -1
7 1 -1 -1 1
Title nonprincipal real characters @\symoperatorsmod\tmspace+.1667em\tmspace+.1667emp are unique
Canonical name NonprincipalRealCharactersmodPAreUnique
Date of creation 2013-03-22 16:34:51
Last modified on 2013-03-22 16:34:51
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 11A25