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That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n. So g is a primitive root modulo n if and only if g is a generator of the multiplicative group of integers modulo n.
The roots of unity modulo n are exactly the integers that are coprime with n. In fact, these integers are roots of unity modulo n by Euler's theorem, and the other integers cannot be roots of unity modulo n, because they are zero divisors modulo n. A primitive root modulo n, is a generator of the group of units of the ring of integers modulo n.
Weisstein, Eric W. "Primitive Root". MathWorld. Web-based tool to interactively compute group tables by John Jones; OEIS sequence A033948 (Numbers that have a primitive root (the multiplicative group modulo n is cyclic)) Numbers n such that the multiplicative group modulo n is the direct product of k cyclic groups:
As Φ 6 (x) = x 2 − x + 1, there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots: +, . As 7 is not a Fermat prime, the seventh roots of unity are the first that require cube roots.
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.
A primitive root modulo m exists if and only if m is equal to 2, 4, p k or 2p k, where p is an odd prime number and k is a positive integer. If a primitive root modulo m exists, then there are exactly φ(φ(m)) such primitive roots, where φ is the Euler's totient function.
In particular, is irreducible if and only if p is a primitive root modulo n, that is, p does not divide n, and its multiplicative order modulo n is (), the degree of . [ 13 ] These results are also true over the p -adic integers , since Hensel's lemma allows lifting a factorization over the field with p elements to a factorization over the p ...
As in the prime-power case, if the conductor equals the modulus the character is primitive, otherwise imprimitive. If imprimitive it is induced from the character with the smaller modulus. For example, χ 15 , 11 {\displaystyle \chi _{15,11}} is induced from χ 3 , 2 {\displaystyle \chi _{3,2}} and χ 15 , 13 {\displaystyle \chi _{15,13}} is ...