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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.
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 ...
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. This is true, because Bézout's identity yields an integer linear combination of k and ℓ equal to gcd ( k , ℓ ) {\displaystyle \gcd(k,\ell )} .
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.
Jacobi's original tables use 10 or –10 or a number with a small power of this form as the primitive root whenever possible, while the second edition uses the smallest possible positive primitive root (Fletcher 1958). The term "canon arithmeticus" is occasionally used to mean any table of indices and powers of primitive roots.
2 is a primitive root mod 9. =) , ... The two orthogonality relations are [25] ... [50] in 1956, and proved in 2017 ...
A non-example is in the ring of integers modulo ; while () and thus is a cube root of unity, + + meaning that it is not a principal cube root of unity. The significance of a root of unity being principal is that it is a necessary condition for the theory of the discrete Fourier transform to work out correctly.
It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity (/ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is