# existence of primitive roots for powers of an odd prime

The following theorem gives a way of finding a primitive root for $p^{k}$, for an odd prime $p$ and $k\geq 1$, given a primitive root of $p$. Recall that every prime has a primitive root.

###### Theorem.

Suppose that $p$ is an odd prime. Then $p^{k}$ also has a primitive root, for all $k\geq 1$. Moreover:

1. 1.

If $g$ is a primitive root of $p$ and $g^{p-1}\neq 1\mod p^{2}$ then $g$ is a primitive root of $p^{2}$. Otherwise, if $g^{p-1}\equiv 1\mod p^{2}$ then $g+p$ is a primitive root of $p^{2}$.

2. 2.

If $k\geq 2$ and $h$ is a primitive root of $p^{k}$ then $h$ is a primitive root of $p^{k+1}$.

Title existence of primitive roots for powers of an odd prime ExistenceOfPrimitiveRootsForPowersOfAnOddPrime 2013-03-22 16:21:01 2013-03-22 16:21:01 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 11-00 EveryPrimeHasAPrimitiveRoot