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existence of primitive roots for powers of an odd prime
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(Theorem)
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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 1 Suppose that $p$ is an odd prime. Then $p^k$ also has a primitive root, for all $k\geq 1$ . Moreover:
- 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 $p^2$ .
- 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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"existence of primitive roots for powers of an odd prime" is owned by alozano.
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Cross-references: every prime has a primitive root, prime, odd, primitive root, theorem
This is version 1 of existence of primitive roots for powers of an odd prime, born on 2006-10-27.
Object id is 8484, canonical name is ExistenceOfPrimitiveRootsForPowersOfAnOddPrime.
Accessed 1236 times total.
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Pending Errata and Addenda
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