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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.
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.
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 ...
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.
Once a primitive kth root of unity x is obtained, every power is a th root of unity, but not necessarily a primitive one. The power x ℓ {\displaystyle x^{\ell }} is a primitive k {\displaystyle k} th root of unity if and only if k {\displaystyle k} and ℓ {\displaystyle \ell } are coprime .
In mathematics, a primitive root may mean: Primitive root modulo n in modular arithmetic Primitive n th root of unity amongst the solutions of z n = 1 in a field
Linear congruence theorem; Method of successive substitution; Chinese remainder theorem; Fermat's little theorem. Proofs of Fermat's little theorem; Fermat quotient; Euler's totient function. Noncototient; Nontotient; Euler's theorem; Wilson's theorem; Primitive root modulo n. Multiplicative order; Discrete logarithm; Quadratic residue. Euler's ...