Artin representations

Definition

Artin representations are a type of representation of Galois groups, particularly used to study the symmetries in number fields. They provide a way to connect algebraic number theory with linear algebra through the action of Galois groups on vector spaces over finite fields or number fields. Artin representations play a crucial role in the Langlands program and have deep implications in understanding the relationship between number theory and geometry.

5 Must Know Facts For Your Next Test

1. Artin representations are typically defined over a number field or a global field, reflecting the Galois group's actions on associated vector spaces.
2. They can be classified into different types based on their properties, such as being irreducible or having specific ramification behaviors at primes.
3. Artin representations often arise in the context of class field theory, where they help to describe abelian extensions of number fields.
4. The study of Artin representations has led to important results like the Artin reciprocity law, which connects number fields with their Galois groups.
5. Understanding Artin representations is essential for exploring L-functions and their conjectural relationships with number theoretic properties.

Review Questions

• How do Artin representations relate to Galois groups and their actions on vector spaces?
  • Artin representations specifically characterize how Galois groups act on vector spaces associated with number fields. They provide a structured way to understand these actions by mapping group elements to linear transformations on these spaces. This relationship helps mathematicians analyze symmetries in algebraic structures, revealing deep insights about both Galois theory and number theory.
• Discuss the significance of Artin representations in relation to the Langlands program and their impact on modern mathematics.
  • Artin representations are significant in the Langlands program as they form a bridge between number theory and representation theory. They provide crucial examples that help in understanding L-functions, which are central to many conjectures within this program. By analyzing Artin representations, mathematicians can explore connections between different areas of mathematics, shedding light on how algebraic structures relate to geometric objects.
• Evaluate how the classification of Artin representations enhances our understanding of class field theory and its applications.
  • The classification of Artin representations enhances our understanding of class field theory by revealing the different ways Galois groups can act on number fields. This classification informs us about the structure of abelian extensions, providing a framework to understand both local and global aspects of number fields. By studying these representations, we can derive important results such as the Artin reciprocity law, thereby gaining deeper insights into the relationships between number theory, algebra, and geometry.

Related terms

Galois Group: A group that describes the symmetries in the roots of a polynomial and provides insights into field extensions.

Linear Representation: A homomorphism from a group to the general linear group, which allows the study of group elements as linear transformations.

Langlands Program: An ambitious set of conjectures linking number theory and representation theory, with far-reaching implications in mathematics.