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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. So g is a primitive root modulo n if and only if g is a generator of the multiplicative group of integers modulo n.
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.
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:
Toggle the table of contents. ... Print/export Download as PDF; Printable version; ... −1 is a primitive root mod 4 ...
Toggle the table of contents. Primitive root. ... Print/export Download as PDF; ... In mathematics, a primitive root may mean: Primitive root modulo n in ...
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.
An extension of K of degree n is a radical extension generated by an nth root of an element of K if and only if it is a Galois extension whose Galois group is a cyclic group of order n. The proof is related to Lagrange resolvents. Let be a primitive nth root of unity (belonging to K).
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