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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.
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.
is a primitive n th root of unity. This formula shows that in the complex plane the n th roots of unity are at the vertices of a regular n-sided polygon inscribed in the unit circle, with one vertex at 1 (see the plots for n = 3 and n = 5 on the right).
In mathematics, a primitive root may mean: Primitive root modulo n in modular arithmetic; Primitive nth root of unity amongst the solutions of z n = 1 in a field; See ...
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 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 .
The other primitive q-th roots of unity are the numbers where (a, q) = 1. Therefore, there are φ( q ) primitive q -th roots of unity. Thus, the Ramanujan sum c q ( n ) is the sum of the n -th powers of the primitive q -th roots of unity.
Primitive element (finite field), an element that generates the multiplicative group of a finite field; Primitive element (lattice), an element in a lattice that is not a positive integer multiple of another element in the lattice; Primitive element (coalgebra), an element X on which the comultiplication Δ has the value Δ(X) = X⊗1 + 1⊗X
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