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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] See modular arithmetic for notation and terminology.
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. There exist primitive roots modulo n if and only if where and are respectively the Carmichael function and Euler's totient function.
A root of unity modulo n is a primitive kth root of unity modulo n for some divisor k of and, conversely, there are primitive kth roots of unity modulo n if and only if k is a divisor of
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Finch, Stephen; Martin, Greg; Sebah, Pascal (2010). "Roots of unity and nullity modulo n" (PDF). Proceedings of the American Mathematical Society. 138 (8): 2729–2743. doi:10.1090/s0002-9939-10-10341-4. Retrieved 2011-02-20.
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g(a;p)=∑n=0p−1ζpan2,{\displaystyle g(a;p)=\sum _{n=0}^{p-1}\zeta _{p}^{an^{2}},} where ζp{\displaystyle \zeta _{p}} is a primitive pth rootofunity, for...
the cubic residue character of α {\displaystyle \alpha } modulo π {\displaystyle \pi } and is denoted by ( α π ) 3 = ω k ≡ α N ( π ) − 1 3 mod π . {\displaystyle...
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element of Z[ζ] that is coprime to a and l {\displaystyle l} and congruent to a rational integer modulo (1–ζ)2. Suppose that ζ is an lth rootofunity for...
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