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.

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- Feb 24th 2010, 05:00 AMNikoBellicprimitive 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.

- Feb 24th 2010, 05:17 AMtonio

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

Tonio