Galois theory

In mathematics, Galois theory was introduced by Évariste Galois. It connects field theory and group theory, making some problems easier to solve. This connection is known as the fundamental theorem of Galois theory.

Galois used this theory to study the roots of polynomials. He could tell which polynomial equations could be solved using simple formulas with integers, nth roots, and basic arithmetic operations. This built on the Abel–Ruffini theorem, which says that equations of degree five or higher cannot always be solved this way.

Galois theory has helped solve old problems, like showing that certain ancient challenges, such as doubling the cube and trisecting the angle, cannot be done in the way they were described. It also helps describe which regular polygons can be constructible.

Although Galois' work was published by Joseph Liouville fourteen years after his death, it took time for other mathematicians to understand and accept it. Later, the ideas of Galois theory were expanded into topics like Galois connections and Grothendieck's Galois theory.

Application to classical problems

Galois theory began because of a big question in math: Can we find a simple formula to solve equations of the fifth degree or higher using just basic math operations and roots? The Abel–Ruffini theorem shows that such a formula does not exist for some equations. 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 and easy 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 started with the study of special math patterns called symmetric functions. These patterns help us understand the roots, 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 have a polynomial, sometimes the roots (solutions) are linked by algebraic equations. For example, two roots might satisfy an equation like A² + 5_B_³ = 7. Galois' theory looks at how we can rearrange, or permute, these roots while keeping all algebraic equations true.

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

For more complex polynomials, like x⁴ − 10x² + 1, the Galois group can have more permutations. In this case, there are four simple permutations that change the signs of square roots, forming a group with four elements.

Modern approach by field theory

In the modern way of looking at things, we start with something called a field extension, which 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, which 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 certain math problems using roots, like square roots or cube roots. This works 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, there are some that 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 don't limit the starting field, this problem is easier, and every finite group can appear as a Galois group. One way to show this is by choosing a field and a finite group, then using some clever math steps to build a new field whose Galois group matches the chosen group.

However, it is still unknown whether 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 that remain unsolved.

Inseparable extensions

Galois theory usually looks at special kinds of extensions called Galois extensions, which are separable. But there are also ways to study more general extensions. For a special type called a purely inseparable extension, we can use a different approach. Instead of using groups, we use something called vector spaces and derivations to understand the relationships between fields.

This helps us see how smaller fields fit inside larger ones in a new way. Important work by mathematicians Jacobson and Brantner & Waldron has helped make this theory stronger and more useful.