primitive root
Given any positive integer , the group of units of the ring is a cyclic group iff is 4, or for any odd positive prime and any non-negative integer . A primitive root is a generator of this group of units when it is cyclic.
Equivalently, one can define the integer to be a primitive root modulo , if the numbers form a reduced residue system modulo .
For example, 2 is a primitive root modulo 5, since
are all with 5 coprime positive integers less than 5.
The generalized Riemann hypothesis implies that every prime number has a primitive root below .
References
Wikipedia, “Primitive root modulo n”
Title | primitive root |
---|---|
Canonical name | PrimitiveRoot |
Date of creation | 2013-03-22 16:04:33 |
Last modified on | 2013-03-22 16:04:33 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 12 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 11-00 |
Synonym | primitive root modulo n |
Synonym | primitive element |
Related topic | MultiplicativeOrderOfAnIntegerModuloM |
Related topic | PrimeResidueClass |
Related topic | UsingPrimitiveRootsAndIndexToSolveCongruences |