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In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. 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.
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:
One can obtain such a root by choosing a () th primitive root of unity (that must exist by definition of λ), named and compute the power () /. If x is a primitive kth root of unity and also a (not necessarily primitive) ℓth root of unity, then k is a divisor of ℓ.
Theorem 2 — For every positive integer n there exists a primitive λ-root modulo n. Moreover, if g is such a root, then there are φ ( λ ( n ) ) {\displaystyle \varphi (\lambda (n))} primitive λ -roots that are congruent to powers of g .
In mathematics, a primitive root may mean: Primitive root modulo n in modular arithmetic Primitive n th root of unity amongst the solutions of z n = 1 in a field
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF( q ) is called a primitive element if it is a primitive ( q − 1) th root of unity in GF( q ) ; this means that each non-zero element of GF( q ) can be written as α i for some natural number i .
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1]
For n = 1, the cyclotomic polynomial is Φ 1 (x) = x − 1 Therefore, the only primitive first root of unity is 1, which is a non-primitive n th root of unity for every n > 1. As Φ 2 (x) = x + 1, the only primitive second (square) root of unity is −1, which is also a non-primitive n th root of unity for every even n > 2.