The notational convenience of the Legendre symbol inspired introduction of several other " symbols " used in algebraic number theory, such as the Hilbert symbol and the Artin symbol.

The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group ; and states that these L-functions are identical to certain Dirichlet L-series or more general series ( that is, certain analogues of the Riemann zeta function ) constructed from Hecke characters.

Langlands attached automorphic L-functions to these automorphic representations, and conjectured that every Artin L-function arising from a finite-dimensional representation of the Galois group of a number field is equal to one arising from an automorphic cuspidal representation.

** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem

The Artin reciprocity law, which is a high level generalization of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field.

In number theory, the Ankeny – Artin – Chowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artin and S. Chowla.

The known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that 10 is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains 37. 395 ..% of the primes.

The problem was partially solved by Emil Artin ( 1924 ; 1927 ; 1930 ) by establishing the Artin reciprocity law which deals with abelian extensions of algebraic number fields.

The Artin reciprocity law, established by Emil Artin in a series of papers ( 1924 ; 1927 ; 1930 ), is a general theorem in number theory that forms a central part of global class field theory.

As Artin reciprocity shows, when G is an abelian group these L-functions have a second description ( as Dirichlet L-functions when K is the rational number field, and as Hecke L-functions in general ).

Since G has an irreducible representation of degree 2, an Artin L-function for such a representation occurs, squared, in the factorisation of the Dedekind zeta-function for such a number field, in a product with the Riemann zeta-function ( for the trivial representation ) and an L-function of Dirichlet's type for the signature representation.

It is the Artin root number.

The Artin root number is, then, either + 1 or − 1.

A stack, as defined above, is an Artin stack if there exists a smooth and surjective representable morphism from ( the stack associated to ) a scheme to X.

In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory.

pointed out that the Artin conjecture follows from strong enough results from the Langlands philosophy, relating to the L-functions associated to automorphic representations for GL ( n ) for all.

The first concerns Artin L-functions for a linear representation of a Galois group ; and the second the frequency with which a given integer a is a primitive root modulo primes p, when a is fixed and p varies.

In general, if K is a Galois extension of Q with Galois group G, its Dedekind zeta function is the Artin L-function of the regular representation of G and hence has a factorization in terms of Artin L-functions of irreducible Artin representations of G.

In accordance with the decomposition of the regular representation into irreducible representations, such a zeta-function splits into a product of Artin L-functions, for each irreducible representation of G. For example, the simplest case is when G is the symmetric group on three letters.

The Artin conjecture on Artin L-functions states that the Artin L-function L ( ρ, s ) of a non-trivial irreducible representation ρ is analytic in the whole complex plane.

More precisely, the Langlands conjectures associate an automorphic representation of the adelic group GL < sub > n </ sub >( A < sub > Q </ sub >) to every n-dimensional irreducible representation of the Galois group, which is a cuspidal representation if the Galois representation is irreducible, such that the Artin L-function of the Galois representation is the same as the automorphic L-function of the automorphic representation.

The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.

For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions.

Artin was also an important expositor of Galois theory, and of the group cohomology approach to class ring theory ( with John Tate ), to mention two theories where his formulations became standard.

Together with his teacher Emil Artin, Tate gave a cohomological treatment of global class field theory, using techniques of group cohomology applied to the idele class group and Galois cohomology.

Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals ( or ideles ) to elements of a Galois group is trivial on a certain subgroup.

A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E, then E / F is a Galois extension, where F is the fixed field of G.

An important theorem of Emil Artin states that for a finite extension E / F, each of the following statements is equivalent to the statement that E / F is Galois:

The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.

In fact, Murty showed that Artin L-functions corresponding to irreducible representations of the Galois group of a solvable extension of the rationals are automorphic as predicted by the Langlands conjectures.

In mathematics, Artin – Schreier theory is a branch of Galois theory, and more specifically is a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p. introduced Artin – Schreier theory for extensions of prime degree p, and generalized it to extensions of prime power degree p < sup > n </ sup >.

Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, and van der Waerden himself on ideals as the main references.

* Artin group

To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold X of dimension at least 2.

* Right-angled Artin groups, a class of groups studied in geometric group theory.

Hecke algebras are quotients of the group rings of Artin braid groups.

* Progress on traditional combinatorial group theory topics, such as the Burnside problem, the study of Coxeter groups and Artin groups, and so on ( the methods used to study these questions currently are often geometric and topological ).

Using the fact that Artin L-functions are holomorphic in a neighbourhood of the line, they showed that for any torsionfree nilpotent group, the function ζ < sub > G </ sub >( s ) is meromorphic in the domain

In mathematics, an Artin group ( or generalized braid group ) is a group with a presentation of the form

The Artin – Mazur zeta function is formally similar to the local zeta function, when a diffeomorphism on a compact manifold replaces the Frobenius mapping for an algebraic variety over a finite field.

His work on the problem of characterising the representable functors in the category of schemes has led to the Artin approximation theorem, in local algebra.