primitive roots

• February 24th 2010, 05:00 AM
NikoBellic
primitive roots
Let g be a primitive root of an odd prime p. Show that -g is also a primitive root of p if and only if p is congruent to 1 mod 4.
• February 24th 2010, 05:17 AM
tonio
Quote:

Originally Posted by NikoBellic
Let g be a primitive root of an odd prime p. Show that -g is also a primitive root of p if and only if p is congruent to 1 mod 4.

-g is a prim. root of p iff $-1=(-g)^\frac{p-1}{2}=(-1)^\frac{p-1}{2}g^\frac{p-1}{2}=(-1)^\frac{p-1}{2}(-1)\Longleftrightarrow (-1)^\frac{p-1}{2}=1$ $\Longleftrightarrow \frac{p-1}{2}=2k\,,\,\,k\in\mathbb{Z}$ $\Longleftrightarrow$ ...

Tonio