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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 number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.
Artin's conjecture on primitive roots The (now proved) conjecture that finite fields are quasi-algebraically closed; see Chevalley–Warning theorem The (now disproved) conjecture that any algebraic form over the p-adics of degree d in more than d 2 variables represents zero: that is, that all p -adic fields are C 2 ; see Ax–Kochen theorem or ...
Theorem 2 — For every positive integer n there exists a primitive λ-root modulo n. Moreover, if g is such a root, then there are φ ( λ ( n ) ) {\displaystyle \varphi (\lambda (n))} primitive λ -roots that are congruent to powers of g .
The primitive element theorem was proved in its modern form by Ernst Steinitz, in an influential article on field theory in 1910, which also contains Steinitz's theorem; [6] Steinitz called the "classical" result Theorem of the primitive elements and his modern version Theorem of the intermediate fields.
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 field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF( q ) is called a primitive element if it is a primitive ( q − 1) th root of unity in GF( q ) ; this means that each non-zero element of GF( q ) can be written as α i for some natural number i .
For a primitive () th root x, the number () / is a primitive th root of unity. If k does not divide λ ( n ) {\displaystyle \lambda (n)} , then there will be no k th roots of unity, at all. Finding multiple primitive k th roots modulo n