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

单词	ExistenceOfPrimitiveRootsForPowersOfAnOddPrime
释义	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. If $g$ is a primitive root of $p$ and $g^{p-1}\eq 1\\mod p^{2}$ 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. If $k\\geq 2$ and $h$ is a primitive root of $p^{k}$ then $h$ is a primitive root of $p^{k+1}$ .
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