primitive roots

**NikoBellic** · February 24th 2010, 05:00 AM

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.

**tonio** · February 24th 2010, 05:17 AM

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

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