nonprincipal real characters are unique
Let be a prime. Then there is a unique nonprincipal real Dirichlet character , given by
Proof. Note first that is obviously a nonprincipal real character . Now, suppose is any nonprincipal real character . Choose some generator, , of . Clearly (since otherwise is principal), and thus . Since , and no lower power of is , it follows that is on exactly elements of and is on exactly elements. However, since , is on each of the squares in . Thus is on squares and on nonsquares, so .
Note that this result is not true if is not prime. For example, the following is a table of Dirichlet characters modulo , all of which are real:
|Title||nonprincipal real characters are unique|
|Date of creation||2013-03-22 16:34:51|
|Last modified on||2013-03-22 16:34:51|
|Last modified by||rm50 (10146)|