Galois Theory

Galois theory is an important part of abstract algebra that provides a deep connection between field theory and group theory. Named after the French mathematician Évariste Galois, this theory is essential for understanding the solutions of polynomial equations in the context of fields and groups. Through Galois theory, you will explore how the symmetries of algebraic equations can be revealed by investigating the structure of the relevant groups.

Historical context

The origins of Galois theory are linked to attempts to solve polynomial equations. For centuries, mathematicians have tried to find solutions for polynomial equations of higher degrees using radicals – expressions of solutions in terms of powers and roots. Solutions for quadratic, cubic and quartic equations were known by the 16th century, but a general formula for fifth-degree equations, or quintiles, proved elusive.

Then came Evariste Galois, who changed the paradigm. Instead of focusing on solving equations directly, Galois used group theory to understand symmetries in the roots of polynomials. His ideas laid the groundwork for determining when a polynomial can be solved using radicals.

Key concepts in Galois theory

1. Area

A field is a set equipped with two operations: addition and multiplication. These operations follow certain rules, which are familiar to you when working with rational numbers, real numbers, etc. Important properties of fields include commutativity, associativity and distributivity of addition and multiplication, the existence of additive and multiplicative identities (0 and 1, respectively), and inverses for every non-zero element.

Some common examples of fields include:

1. The set of rational numbers (( mathbb{Q} )) 2. The set of real numbers (( mathbb{R} )) 3. The set of complex numbers (( mathbb{C} ))

2. Polynomial equations and expansions

Consider a polynomial ( f(x) ) over a field ( F ). Solving ( f(x) = 0 ) means finding elements in a larger field ( E ) that satisfy the equation. Such a field ( E ) is called a field extension of ( F ), denoted as ( E/F ). Field extensions are important in studying the roots of polynomials.

3. Automorphisms and Galois groups

An automorphism of a field extension ( E/F ) is a bijective (one-to-one and onto) map from ( E ) to itself that preserves the field operations and leaves every element of ( F ) unchanged. The set of all automorphisms of ( E/F ) forms a group, known as the Galois group.

For example, consider the field extension ( mathbb{C}/mathbb{R} ). In this case, complex conjugation is a remarkable automorphism, and the Galois group consists of two elements: the identity map and the conjugation map.

4. The fundamental theorem of Galois theory

One of the greatest achievements of Galois theory is the fundamental theorem of Galois theory. It provides a binary correspondence between subfields of a Galois extension and subgroups of its Galois group. This theorem shows how the structure of intermediate fields of a field extension reflects the subgroup structure of the Galois group.

Detailed exploration of Galois theory

Polynomials and fields

At the heart of understanding Galois theory is the concept of polynomial equations. Consider a polynomial of degree ( n ):

f(x) = a_n x^n + a_{n-1} x^{n-1} + ldots + a_1 x + a_0

where the coefficients ( a_i ) belong to a field ( F ). The solutions of this polynomial may not lie within the field ( F ), but they may exist in a larger field ( E ), which is an extension of ( F ).

Take for example the polynomial ( x^2 - 2 = 0 ). The solutions are ( sqrt{2} ) and ( -sqrt{2} ), neither of which is in the set of rational numbers ( mathbb{Q} ), but both can be found in the real number field ( mathbb{R} ).

Construction of field extensions

A field extension ( E/F ) is obtained by adding new elements to ( F ). These new elements are typically the roots of polynomials with coefficients in ( F ). For example, if we add ( sqrt{5} ) to ( mathbb{Q} ), we get a field extension ( mathbb{Q}(sqrt{5}) ). This field extension contains all numbers of the form ( a + bsqrt{5} ), where ( a, b ) are rational numbers.

You can visualize it as follows:

[ begin{array}{cccc} & & mathbb{Q}(sqrt{5}) & \ & nearrow & & searrow \ mathbb{Q} & & & & mathbb{Q}(sqrt{2}) \ end{array} ]

This diagram represents a simple field extension with ( mathbb{Q}(sqrt{5}) ) and ( mathbb{Q}(sqrt{2}) ) as extended fields of ( mathbb{Q} ).

Understanding Galois groups

To understand symmetries in the roots of polynomials, examine the concept of the Galois group of a field extension. The Galois group describes how the roots can be 'ordered' without changing the original relations. Consider the polynomial ( x^2 - 2 ). The field extension ( mathbb{Q}(sqrt{2})/mathbb{Q} ) has two automorphisms: one is the identity (which maps each element to itself), and another which maps ( sqrt{2} ) to ( -sqrt{2} ).

These automorphisms form a group, denoted ( text{Gal}(mathbb{Q}(sqrt{2})/mathbb{Q}) ). This is a simple group with just two elements: the identity ( tau_0 ) and the map ( tau_1 ) switching ( sqrt{2} ) and ( -sqrt{2} ). Here is a simple table of these elements:

[ begin{array}{c|cc} & sqrt{2} & -sqrt{2} \ hline tau_0 & sqrt{2} & -sqrt{2} \ tau_1 & -sqrt{2} & sqrt{2} \ end{array} ]

The power of the fundamental theorem of Galois theory

The fundamental theorem of Galois theory forms a bridge between field extensions and group theory. A key principle of this theorem is that for a given Galois extension ( E/F ), there is a one-to-one correspondence between intermediate subfields of ( F subseteq K subseteq E ) and subgroups of the Galois group ( text{Gal}(E/F) ).

Consider this correspondence with an example of a field extension hierarchy. Suppose we associate two square roots, ( sqrt{2} ) and ( sqrt{3} ) to ( mathbb{Q} ). Here is a diagram of the structure:

[ begin{array}{cccc} & & mathbb{Q}(sqrt{2},sqrt{3}) & \ & nearrow & & nwarrow \ mathbb{Q}(sqrt{2}) & & & & mathbb{Q}(sqrt{3}) \ & nwarrow & & nearrow \ & & mathbb{Q} & \ end{array} ]

Each line represents a field extension where:

• (mathbb{Q}(sqrt{2},sqrt{3})) over (mathbb{Q}) is a complete Galois extension.
• (mathbb{Q}(sqrt{2})) and (mathbb{Q}(sqrt{3})) are intermediate expansions.

The corresponding Galois group relations can be represented as:

• Every subgroup of (text{Gal}(mathbb{Q}(sqrt{2}, sqrt{3})/mathbb{Q})) correlates with a subfield.
• For each subgroup, its definite region corresponds to a particular intermediate region.

Applications of Galois theory

Solving polynomials by radicals

An important result of applying Galois theory is to determine whether a polynomial is solvable by radicals, meaning that its roots can be expressed as a combination of basic arithmetic operations and nth roots. Using the theorem, we can establish that a polynomial is solvable by radicals when its associated Galois group is a soluble group, a concept from group theory.

For example, consider the quinary equation:

f(x) = x^5 - x + 1 = 0

Galois theory proves that it is insoluble by radicals, because the Galois group is as large and complex as (text{S}_5), the symmetric group on five elements, which is insoluble.

Geometric constructions

Galois theory can also be applied to classic problems on geometric constructions with ruler and compass. By understanding what lengths can be constructed, Galois theory helps prove the limits of constructions, such as squaring a circle, doubling a cube, or bisecting an angle.

For example, it is possible to construct a regular polygon with 17 sides using only a ruler and a compass, a fact verified by means of corresponding area expansions, which are fundamental expansions.

Conclusion

Galois theory is a testament to the beauty and interplay between algebraic structures. It enriches the understanding of polynomials by highlighting their solvability and goes beyond traditional algebra by connecting fields to groups. Galois' groundbreaking insights have played a key role in mathematical progress and discovery, demonstrating the unmatched beauty and power of pure mathematics.

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