An Artin representation is a homomorphism from the absolute Galois group of a number field to the general linear group over the complex numbers, often used to study the relationship between field extensions and linear algebra. Named after Emil Artin, these representations play a crucial role in modern number theory, particularly in the Langlands program and the study of L-functions.

Artin Representation

The Absolute Galois Group is a fundamental concept in algebraic number theory and algebraic geometry, representing the group of all field automorphisms of the algebraic closure of a field that fix the field itself. It encapsulates the symmetries of the roots of polynomials over the field, providing deep insights into the structure and solvability of equations.

Absolute Galois Group

A number field is a finite degree field extension of the rational numbers, providing a rich structure for studying algebraic numbers and their properties. Number fields are fundamental in algebraic number theory, as they generalize the rational numbers and allow for the exploration of Diophantine equations and prime factorization in more complex settings.

Number Field

General Linear Group

Complex numbers extend the real numbers by including the imaginary unit 'i', which is defined as the square root of -1, allowing for the representation of numbers in the form a + bi, where a and b are real numbers. This extension enables solutions to polynomial equations that have no real solutions and facilitates advanced mathematical and engineering applications, particularly in fields like signal processing and quantum mechanics.

Complex Numbers

Field extensions are a fundamental concept in abstract algebra that involve expanding a given field to a larger field in which more equations can be solved. They are essential in understanding the structure of fields and play a crucial role in areas such as Galois theory, which connects field theory with group theory to solve polynomial equations.

Field Extensions

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces, focusing on the study of lines, planes, and subspaces. It is fundamental in various scientific fields, providing tools for solving systems of linear equations, performing transformations, and analyzing vector spaces and matrices.

Linear Algebra

Emil Artin

Modern number theory

The Langlands program is a set of conjectures and theories that connect number theory and representation theory, proposing a grand unifying framework for understanding the relationships between Galois groups and automorphic forms. It has profound implications across mathematics, linking areas such as algebraic geometry, arithmetic geometry, and mathematical physics, and has led to significant advancements in these fields.

Langlands Program

L-functions are complex analytic functions that encode number-theoretic information and are central to many deep results in mathematics, including the proof of the Prime Number Theorem and the Birch and Swinnerton-Dyer Conjecture. They are often associated with automorphic forms and have applications in fields such as algebraic geometry, representation theory, and mathematical physics.

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