Galois theory

In mathematics, Galois theory was introduced by Évariste Galois. It connects field theory and group theory, helping us solve some problems more easily. This important link is called the fundamental theorem of Galois theory.

Galois used this theory to study the solutions of polynomials. He figured out which polynomial equations can be solved using simple math with integers, nth roots, and basic arithmetic operations. This work relates to the Abel–Ruffini theorem, which shows that some equations of the fifth degree or higher cannot always be solved this way.

Galois theory has helped solve old puzzles, like proving that some ancient challenges, such as doubling the cube and trisecting the angle, cannot be done the way they were described. It also explains which regular polygons can be constructible.

Even though Galois' work was shared by Joseph Liouville long after he passed away, it took time for other mathematicians to understand and accept it. Later, these ideas grew into new areas like Galois connections and Grothendieck's Galois theory.

Application to classical problems

Galois theory started because of a big question in math: Can we find a simple way to solve equations of the fifth degree or higher using just basic math and roots? The Abel–Ruffini theorem shows that for some equations, this is not possible. Galois theory explains why we can solve equations of degree four or lower this way, but not most equations of degree five or higher. It also helps us understand which equations can be solved using a clear method.

This theory also helps us understand problems in geometry that can be solved using just a compass and straightedge. It explains which shapes we can draw perfectly, why we can’t always split angles into three equal parts, and why we can’t always double the size of a cube using these tools.

History

Galois theory began with the study of special math patterns called symmetric functions. These patterns help us understand the answers, or solutions, of equations.

Many smart people worked on solving equations with many answers. Some, like Évariste Galois, found new ways to see if these equations could be solved easily. Galois showed that by looking at how the answers could be rearranged, we could tell if there was a simple way to solve the equation. This idea connected two big parts of math: working with numbers and working with groups of actions.

Permutation group approach

When we solve a math problem, sometimes the answers are connected by special rules. For example, two answers might fit into an equation like A² + 5_B_³ = 7.

Galois' theory studies how we can rearrange these answers, called permutations, while still following all the rules.

These permutations form a group, called the Galois group. For a simple quadratic equation like x² − 4x + 1 = 0, the answers are A = 2 + √3 and B = 2 − √3. Switching A and B still keeps equations like A + B = 4 and A × B = 1 true. The Galois group here has two ways to arrange the answers: leaving them as they are, or switching A and B.

For harder problems, like x⁴ − 10x² + 1, the Galois group can have more arrangements. In this example, there are four simple ways to change the answers by flipping the signs of square roots, making a group with four parts.

Modern approach by field theory

In the modern way of looking at things, we start with something called a field extension. This is like connecting two sets of numbers.

We then look at special changes we can make to the bigger set of numbers that don’t change the smaller set.

This new way has many benefits. It makes a big rule in Galois theory easier to understand. It also lets us use different number sets besides the simple ones we usually think about. This is important in many parts of math. This way also helps us study bigger connections between number sets and recognize when different equations end up creating the same number sets.

Solvable groups and solution by radicals

The idea of a solvable group in group theory helps us understand when we can solve some math problems using roots, like square roots or cube roots. This is possible when the group linked to the problem has a special property called solvability.

One big success of Galois Theory was showing that for problems with more than four parts, some cannot be solved just by using roots, products, and sums of known numbers. This was also shown earlier by Niels Henrik Abel and is known as the Abel–Ruffini theorem.

A non-solvable quintic example

Van der Waerden mentions the equation f(x) = x⁵ − x − 1 as an example. This equation cannot be solved using simple math rules. Its properties show it is linked to a complex group, making it one of the simplest examples of a problem that cannot be solved with roots alone. Serge Lang and Emil Artin liked using this example to explain the idea.

Inverse Galois problem

Main article: Inverse Galois problem

The inverse Galois problem is about finding a special kind of math structure called a field extension that has a certain Galois group.

When we start with any field, this problem is easier, and every finite group can be a Galois group. We can show this by choosing a field and a finite group, then using some math steps to build a new field whose Galois group matches the chosen group.

But, we still do not know if every finite group can be the Galois group of a field extension of the rational numbers Q. Some experts have shown this works for many types of groups, but there are still some cases we do not know about.

Inseparable extensions

Galois theory usually studies special kinds of extensions called Galois extensions. But there are also ways to study more general extensions. For a special type called a purely inseparable extension, we can use a different method. We use vector spaces and derivations to understand how smaller fields fit inside larger ones.

Important work by mathematicians Jacobson and Brantner & Waldron has helped make this theory stronger and more useful.