Artin representations

Definition

Artin representations are a specific type of linear representation of a Galois group associated with a number field. They provide a bridge between number theory and representation theory by allowing us to study how Galois groups act on vector spaces, facilitating insights into the structure of field extensions and the solutions to polynomial equations.

5 Must Know Facts For Your Next Test

1. Artin representations are named after Emil Artin, who developed the concept in the early 20th century.
2. These representations are particularly important in understanding local and global fields, connecting them to class field theory.
3. Artin representations can be classified into different types based on their properties, such as being unramified or tamely ramified.
4. The dimension of an Artin representation is finite, reflecting the fact that it can be represented by matrices over a base field.
5. Artin L-functions, which are generated from these representations, play a crucial role in modern number theory and are tied to deep conjectures like the Langlands program.

Review Questions

• How do Artin representations connect the concepts of Galois groups and vector spaces?
  • Artin representations serve as a link between Galois groups and vector spaces by providing a way to express how the elements of a Galois group act linearly on a vector space. This connection allows mathematicians to analyze the structure and properties of field extensions by examining these linear actions. In essence, Artin representations translate abstract group actions into more tangible mathematical objects that can be studied using linear algebra techniques.
• What is the significance of classifying Artin representations based on their ramification properties?
  • Classifying Artin representations by their ramification properties is significant because it helps to understand how these representations behave in different local contexts. For instance, unramified representations correspond to behaviors where the Galois group acts without introducing new ramification points, leading to simpler structures. In contrast, ramified representations indicate more complex interactions that can influence important number-theoretic phenomena like the behavior of primes in extensions. This classification provides insights into the broader relationship between number theory and representation theory.
• Evaluate the implications of Artin L-functions derived from Artin representations in relation to number theory's major conjectures.
  • The implications of Artin L-functions derived from Artin representations are profound, as these functions not only encapsulate important number-theoretic information but also connect to major conjectures like the Langlands program. These L-functions encode data about the distribution of prime numbers and have deep links with other areas such as modular forms and automorphic representations. Understanding their properties and relationships helps mathematicians explore questions around rational solutions to polynomial equations and forms the backbone for many modern results in algebraic number theory.

Related terms

Galois Group: The group of symmetries (automorphisms) of a field extension that describes how the roots of polynomials behave under these transformations.

Linear Representation: A homomorphism from a group to the general linear group GL(V), where V is a vector space, capturing how group elements can be represented as linear transformations.

Number Field: A finite degree field extension of the rational numbers, which can be thought of as containing algebraic numbers derived from polynomials with rational coefficients.