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If = is an odd number (/) is cyclic of order (); a generator is called a primitive root mod . [14] Let g q {\displaystyle g_{q}} be a primitive root and for ( a , q ) = 1 {\displaystyle (a,q)=1} define the function ν q ( a ) {\displaystyle \nu _{q}(a)} (the index of a {\displaystyle a} ) by
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.
In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) (). If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. [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.
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 ...
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:
In any ring, if n is a power of 2, then any n/2-th root of −1 is a principal n-th root of unity. A non-example is 3 {\displaystyle 3} in the ring of integers modulo 26 {\displaystyle 26} ; while 3 3 ≡ 1 ( mod 26 ) {\displaystyle 3^{3}\equiv 1{\pmod {26}}} and thus 3 {\displaystyle 3} is a cube root of unity , 1 + 3 + 3 2 ≡ 13 ( mod 26 ...
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. It is a fact [3] that the powers of ζ q are precisely the primitive roots for all the divisors of q. Example. Let ...