A Layman's Introduction to Galois Theory: Understanding the Symmetries of Numbers and Field Extensions

Galois theory is a branch of algebraic mathematics that studies field extensions and their symmetries. It was developed by the French mathematician Évariste Galois in the 19th century, and it has since become an important area of study in algebra and number theory.

The main focus of Galois's theory is understanding the relationship between a field extension and the group of automorphisms of that extension, known as the Galois group. The Galois group encodes information about the symmetries of the extension and can reveal important properties of the extension, such as its degree, its solvability by radicals, and its algebraic independence from the base field.

The fundamental theorem of Galois, which establishes a one-to-one correspondence between subfields of a Galois extension and subgroups of its Galois group, is one of the main findings of Galois theory. We can use this correspondence to comprehend the Galois group's structure and use it to study the extension.

Interdisciplinary Relation:

Numerous branches of mathematics and science, such as number theory, algebraic geometry, physics, and cryptography, all make use of the Galois theory.

Galois's theory, for instance, is used in cryptography to create safe encryption algorithms and is crucial to the demonstration of Fermat's Last Theorem.

Background Knowledge:

Let's begin with some fundamental definitions:

1. A field is a mathematical object that satisfies certain axioms, such as the existence of addition, subtraction, multiplication, and division operations. Real numbers, complex numbers, and rational numbers are a few examples of fields.
2. A larger field that encompasses a smaller field is referred to as a field extension. Real numbers are an extension of the field of complex numbers, for instance.
3. A function that preserves the field operations and takes the identity element to itself is said to be an automorphism of the field. The complex conjugation map, for instance, is an automorphism of the complex number field.

With these definitions in mind, let's move on to Galois's theory proper. The main idea of Galois's theory is to study the relationship between a field extension and its automorphisms.

Given a field extension K ⊆ L, we can consider the set of automorphisms of L that fix K pointwise. This set is denoted by Gal(L/K) and is called the Galois group of the extension. 

"The Galois group is a group under the composition of functions, which means it satisfies certain axioms such as the existence of an identity element and inverse elements."

The Galois group encodes information about the symmetries of the extension. For example, if L is a finite extension of K, then the degree of the extension [L: K] is equal to the order of the Galois group |Gal(L/K)|. This means that the Galois group tells us how many "symmetries" the extension has.

The fundamental theorem of Galois establishes a correspondence between subfields of L that contain K and subgroups of Gal(L/K). This correspondence is given by the following formula:

{subfields of L that contain K} <-> {subgroups of Gal(L/K)}

This correspondence allows us to use group theory to study field extensions. For example, we can classify all subfields of a Galois extension by classifying all subgroups of its Galois group.

Another important concept in Galois's theory is that of a normal extension. An extension K ⊆ L is said to be normal if it is invariant under all automorphisms of L that fix K. In other words, if α is an automorphism of L that fixes K, then α(L) = L. Normal extensions are important because they have a particularly nice Galois group structure. In particular, if K ⊆ L is a normal extension, then the Galois group Gal(L/K) is a transitive group of automorphisms, which means it acts transitively on the roots of any polynomial in K[x] that has a root in L.

Galois's theory helps us understand the relationships between different kinds of numbers, like real numbers, complex numbers, and more complicated numbers called field extensions. It does this by looking at the symmetries of these numbers.

Symmetry is a very important concept in mathematics. For example, think about a square. If you rotate the square by 90 degrees, it still looks the same. This is a symmetry of the square. Similarly, when we look at numbers, some of them have symmetries too.

The Galois group of a field extension tells us about the symmetries of the numbers in that extension. For example, if we have a field extension of the real numbers that includes some complex numbers, the Galois group tells us about the different ways we can manipulate those complex numbers while keeping the real numbers fixed.

The Galois group can tell us how many symmetries the numbers in the extension have. We can use this information to understand how the numbers in the extension behave, and what kinds of equations we can solve with them.

The Galois group can be related to the extension's subfields using the fundamental theorem of Galois.

We can better comprehend the extension and its symmetries by looking at the subfields.

In conclusion, the Galois theory is a useful tool for understanding the symmetries of various types of numbers and how they relate to the field extensions that contain them.

Numerous branches of mathematics and science have significant applications for it.

Galois's theory has important applications in various areas of mathematics and science. It is used in algebraic number theory to study the behavior of prime numbers in algebraic extensions of rational numbers, and in physics to study the symmetries of physical systems.

If you want to learn further, I would recommend you to check out the book "Galois Theory by Ian Stewart".