Galois Theory

Galois theory, named after the mathematician Évariste Galois, is a branch of abstract algebra that provides a deep connection between field theory and group theory. It enables us to understand the symmetries within the roots of polynomial equations. The main purpose of Galois theory is to provide a comprehensive framework for determining when a polynomial equation can be solved by radicals, that is, solutions expressed using addition, subtraction, multiplication, division, and root extraction.

Field

Before delving deeper into Galois theory, it is important to understand the concept of a field. A field is a set equipped with two operations: addition and multiplication, which satisfy certain conditions, much like familiar number systems such as rational numbers, real numbers, and complex numbers.

Formally, the field F is a set with two operations, commonly called addition and multiplication, for which the following conditions hold:

• Associativity: For any a, b, c ∈ F, we have (a + b) + c = a + (b + c) and (a cdot b) cdot c = a cdot (b cdot c).
• Commutativity: For any a, b ∈ F, we have a + b = b + a and a cdot b = b cdot a.
• Identity element: There exist elements 0 and 1 in F such that for every element a ∈ F, a + 0 = a and a cdot 1 = a.
• Inverse: For every a ∈ F, there exists an element -a ∈ F (an additive inverse) such that a + (-a) = 0 Moreover, for every a ≠ 0, there exists an element a^(-1) ∈ F (a multiplicative inverse) such that a cdot a^(-1) = 1.
• Distributivity: For all a, b, c ∈ F, we have a cdot (b + c) = a cdot b + a cdot c.

Vectors and polynomials

Polynomials play an important role in Galois theory. A polynomial is an expression of the form:

p(x) = a_n cdot x^n + a_{n-1} cdot x^{n-1} + ... + a_1 cdot x + a_0

where a_i are coefficients from the field F and x is a variable. A fundamental problem in algebra is solving polynomial equations. For example:

f(x) = x^2 - 4x + 4

The root in polynomial factors form is x = 2 (x - 2)^2.

Field extensions

Field extensions are fundamental to understanding Galois theory. If E is a field containing a subfield F, then E is an extension of F, written E/F.

F = mathbb{Q}, which is the field of rational numbers, and E = mathbb{Q}(sqrt{2}), which contains all numbers of the form a + bsqrt{2}, where a, b ∈ mathbb{Q} Clearly, E contains F since it contains every rational number (take b = 0).

Construction of the field

The simplest way to construct a field extension is to add roots of a polynomial. For example, given a polynomial f(x) = x^2 - 2 with a root of sqrt{2}, we can construct an extension of mathbb{Q}(sqrt{2}) to mathbb{Q}, resulting in mathbb{Q}(sqrt{2})

Visual example of field extensions

In this illustration, F is the base field, and E is an extension field. The set E contains F, and the extension E/F shows that E is formed by taking all the elements of F and adding additional elements.

Automorphisms and Galois groups

To understand Galois theory in depth, it is important to understand the concept of automorphism. An automorphism is a bijective map from a field to itself that respects the field operations. If F is a field, then the automorphism sigma: F to F must satisfy:

• sigma(a + b) = sigma(a) + sigma(b) for any a, b ∈ F
• sigma(a cdot b) = sigma(a) cdot sigma(b) for any a, b ∈ F

For a field extension E/F, an automorphism of E that fixes every element of F is called a field automorphism. The set of all such automorphisms forms a group under function composition, called the Galois group of the extension, denoted Gal(E/F).

Exploring a simple example

Consider the polynomial x^2 - 2 over mathbb{Q}. The roots are sqrt{2} and -sqrt{2}. We form the extension mathbb{Q}(sqrt{2}) consisting of all numbers a + bsqrt{2} where a, b ∈ mathbb{Q}.

The Galois group of this extension contains two automorphisms:

• The identity automorphism that maps every element onto itself, sigma_1(x) = x.
• The automorphism sigma_2 that sends sqrt{2} to -sqrt{2} effectively switches between the origins.

Fundamental theorem of Galois theory

One of the most fundamental results in Galois theory is the fundamental theorem of Galois theory. It establishes a deep connection between field extensions and their associated Galois groups.

The statement is as follows:

Let E/F be a Galois extension whose Galois group G = Gal(E/F). There exists a one-to-one correspondence between subgroups of G and intermediate fields K such that for every subgroup H of G F ⊆ K ⊆ E, the corresponding intermediate field is the constant field of H, defined as:

K = { x ∈ E | sigma(x) = x text{ for all } sigma ∈ H }

This correspondence reveals the underlying symmetry of polynomial solutions and their algebraic structures. It allows us to conclude whether certain polynomial equations can be solved by radicals based on the properties of their Galois groups.

Understanding the basic principles

I K F

Concluding reflection

Galois theory elegantly addresses the puzzle of polynomial equations that can be solved in radicals by analyzing the symmetries of their roots. Its rich interconnection between fields and groups forms a major pillar in abstract algebra, deeply influencing many areas of mathematics, including number theory and geometry. Through understanding and manipulating field extensions, polynomial roots, and their symmetries via Galois groups, Galois theory provides a bridge between seemingly disparate mathematical concepts, offering insight into the deep structures that govern algebraic solutions.

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