Search results
Results From The WOW.Com Content Network
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.
For example, if =, every positive integer less than 17 is a 16th root of unity modulo 17, and the integers that are primitive 16th roots of unity modulo 17 are exactly those such that / (). Finding a primitive k th root of unity modulo n
Moreover, there exist more informative radical expressions for n th roots of unity with the additional property [12] that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive n th root of unity.
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 this case, a primitive element is also called a primitive root modulo q. For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup {2, 4, 1} of order 3; however, 3 is a primitive element of GF(7). The minimal polynomial of a primitive element is a primitive polynomial.
As the 3rd and the 7th roots of unity belong to GF(4) and GF(8), respectively, the 54 generators are primitive n th roots of unity for some n in {9, 21, 63}. Euler's totient function shows that there are 6 primitive 9 th roots of unity, 12 {\displaystyle 12} primitive 21 {\displaystyle 21} st roots of unity, and 36 {\displaystyle 36} primitive ...
The other four elements of the multiplicative group modulo 15, namely 1, 4 (which satisfies ), 11, and 14, are not primitive λ-roots modulo 15. For a contrasting example, if n = 9, then () = = and (()) =. There are two primitive λ-roots modulo 9, namely 2 and 5, each of which is congruent to the fifth power of the other. They are also both ...
An nth root of unity is a complex number whose nth power is 1, a root of the polynomial x n − 1. The set of all nth roots of unity forms a cyclic group of order n under multiplication. [1] The generators of this cyclic group are the nth primitive roots of unity; they are the roots of the nth cyclotomic polynomial.