Galois Representations

In algebraic number theory and arithmetic geometry, the study of Galois representations presents a deep connection between field extensions and linear algebra. It is part of a vibrant field touching number theory, algebra, and geometry. Understanding Galois representations helps mathematicians tackle complex problems involving polynomials, numbers, and symmetry.

Basics of Galois theory

Before delving deeper into Galois representations, we need some understanding of Galois groups and field extensions. A Galois group is associated with a polynomial or field extension and consists of the algebraic symmetries—automorphisms—of that structure that map fields onto themselves while preserving addition, multiplication, and rational coefficients.

Consider a polynomial equation:

x^5 - 2 = 0

Finding its roots gives the splitting field, which is the smallest extension field where the polynomial can be split exactly. The Galois group of this field tells how the roots can be permuted.

Field extensions and automorphisms

The concept of a field extension is fundamental. If you have a field K and a larger field L containing K, then L is called an extension of K An automorphism is a self-isomorphism of a field; it is a bijective map that preserves the field operation.

The Galois group of an extension L/K, denoted Gal(L/K), is the group of all field automorphisms of L that stabilize K

For example, field extensions:

Q(√2)

There is an automorphism that maps √2 to -√2. This automorphism forms a Galois group with two elements - the identity and itself.

Understanding Galois representation

The Galois representation is a homomorphism from the Galois group of a field extension to a general linear group. To understand in detail, consider a Galois group Gal(L/K) and a vector space V over a field F The representation of Gal(L/K) is a group homomorphism:

ρ: Gal(L/K) → GL(V)

where GL(V) denotes the group of invertible linear transformations of the vector space V

Example: cyclotomic field

To understand this concept, consider a cyclotomic field. A cyclotomic field is generated by adjoining a primitive root of unity to a rational number Q The nth roots of unity generate such a field Q(ζ_n).

x^n - 1 = 0

The Galois group of this cyclotomic field can be represented, say for n = 3, as a subgroup of the permutation group of the roots: {1, ζ, ζ^2}.

Visual example

Consider a more visual example with roots of unity:

The three points on the circle represent the cube roots of unity. The Galois group permutes these roots, and each permutation can be expressed in terms of matrices acting on the vectors, which represent the group.

Linear algebra and Galois representations

Representations of Galois groups are deeply connected with linear algebra. When we set up the representations, each element of the Galois group can be thought of as a matrix. These matrices act on the vector space to reflect various symmetries of roots and field structures.

Example: 2D representation

Consider a two-dimensional vector space over the field F and the Galois group G of order 2, which can be represented as:

G = {e, σ}, where e is the identity and σ is a non-trivial automorphism.

A representation shall specify the following:

ρ(e) = I = |1 0| |0 1| ρ(σ) = A = |0 1| |1 0|

Here, I is the identity matrix and A denotes a nontrivial permutation of the coordinates in the vector space.

Application of the Galois representation

Widespread applications of Galois representations are found in the study of modular forms, elliptic curves, and number fields.

Elliptic curve

An elliptic curve is described by an equation:

y^2 = x^3 + ax + b

Its Galois representations trace the action of the Galois group on points defined on the curve, and reveal information about its rational points and rank.

Modular form

Modular forms are analytic functions that are invariant under a particular group of transformations. Galois representations associated with modular forms lead to breakthroughs such as the proof of Fermat's Last Theorem by Andrew Wiles.

Summary

Galois representations are at the heart of much of modern mathematics, connecting such diverse fields as algebra and number theory. By understanding the algebraic symmetries of field extensions and their corresponding matrix representations, mathematicians gain deeper insights into the fundamental nature of numbers and shapes.

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