Galois group

In Galois theory, a part of abstract algebra, a Galois group helps us understand how some number systems grow from smaller ones. Each piece of a Galois group is a special kind of change that moves parts of the larger system but leaves the smaller system the same.

This link between number systems and groups lets us use ideas from group theory to solve problems about number systems. For example, it helps us understand the solutions to certain kinds of equations called quintic polynomials. The whole area of studying these connections is named Galois theory after Évariste Galois, who first discovered them.

Informal description

A field is a set of numbers, like the rational numbers or the real numbers, where you can add, subtract, multiply, and divide. We can make bigger fields from smaller ones. For example, the rational numbers can be made bigger to include the real numbers, and the real numbers can be made even bigger to include complex numbers.

Sometimes, we can't solve an equation with numbers in a smaller field, but we can in a bigger one. For example, the equation x² = 2 has no solution in the rational numbers, but it does in the real numbers.

We can use a special kind of math called group theory to study these bigger fields. The Galois group helps us understand the extra structure added when we make a bigger field. It looks at changes that affect the bigger field but leave the smaller one unchanged. For example, switching a complex number to its "mirror" form, called complex conjugation, leaves real numbers unchanged but changes most complex numbers. This switch is part of the Galois group for complex numbers over real numbers.

Definition

Imagine you have two sets of numbers, where one set is a bigger version of the other. We can call these "fields" in math.

A Galois group is a way to show how the bigger set of numbers relates back to the smaller one. It looks at special rules (called automorphisms) that change the bigger set but keep the smaller set the same. When we put all these rules together, they form a group — a collection of actions that can be mixed in special ways.

This idea helps mathematicians solve tough problems, like finding what the answers to some equations might be.

Structure of Galois groups

Fundamental theorem of Galois theory

The fundamental theorem of Galois theory shows a special link between two math ideas: fields and groups. It says that for some types of field extensions, there is a clear matching between parts of the smaller field and the subgroups of a group linked to the larger field. This matching helps mathematicians solve field problems using group theory tools.

When a smaller group is a special type called a "normal subgroup," the theorem helps us understand how the larger group's structure connects to the smaller field's structure. This is useful for studying solutions to certain equations.

Lattice structure

When we combine two Galois extensions — special types of field extensions — their Galois groups relate in a clear way. If the two original fields only share the base field and nothing else, then the Galois group of the combined field links directly to the product of the two individual Galois groups.

Inducting

This idea can be used for many field extensions together. If we have several Galois extensions where each new extension only shares the base field with all the previous ones, then the Galois group of the overall combined field matches the product of all the individual Galois groups.

Examples

In these examples, F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F(a) means adding an element a to the field F to create a new field.

Computational tools

Cardinality of the Galois group and the degree of the field extension

A key idea in finding the Galois group of a field extension is that the size of the Galois group matches the degree of the extension. This means the number of different ways you can rearrange the elements while keeping the basic structure the same is equal to how much bigger the new field is compared to the original.

Eisenstein's criterion

A useful method for finding the Galois group of a polynomial comes from Eisenstein's criterion. If a polynomial can be broken down into smaller, irreducibility parts, the Galois group of the whole polynomial includes the Galois groups of each part.

Trivial group

The Galois group Gal(F/F) is the trivial group with just one element, the identity map.

Another example of a trivial Galois group is Aut(R/Q). Any transformation of the real numbers must keep their order, so it can only be the identity map.

Consider the field K = Q(∛2). The group Aut(K/Q) only contains the identity map because K is not a normal extension.

Finite abelian groups

The Galois group Gal(C/R) has two elements: the identity map and complex conjugation.

Quadratic extensions

The degree two extension Q(√2)/Q has a Galois group with two elements: the identity map and a map that swaps √2 and −√2. This idea extends to other prime numbers.

Product of quadratic extensions

For different prime numbers p1, ..., pk, the Galois group of Q(√p1, ..., √pk)/Q is a product of the Galois groups of each individual extension.

Cyclotomic extensions

Cyclotomic polynomials help create examples of Galois groups. These polynomials have a special form and their splitting fields lead to interesting Galois groups.

The splitting field of x³ − 2 over Q has the Galois group of the cyclic group of order 6.

Quaternion group

The Quaternion group appears as the Galois group of a certain field extension of Q.

Symmetric group of prime order

If a polynomial of prime degree has exactly two non-real roots, its Galois group is the full symmetric group.

For example, the polynomial f(x) = x⁵ − 4x + 2 has three real roots and two complex roots, so its Galois group is S₅.

Comparing Galois groups of field extensions of global fields

When studying extensions of global fields like Q(∛5, ζ₅)/Q, you can compare Galois groups using valuations. This helps build Galois groups of local fields from global ones.

Infinite groups

A field extension with infinitely many automorphisms is Aut(C/Q), which includes all algebraic extensions of Q inside C.

The absolute Galois group is an infinite group formed by taking the limit of all finite Galois extensions of a field. It’s a topological group.

Another example is the field extension Q(√2, √3, √5, …)/Q, which contains the square roots of all positive primes. Its Galois group is a product of Z/2Z for each prime.

Properties

A Galois extension is special because it follows the fundamental theorem of Galois theory. This theorem says that certain groups of the extension match up with the middle steps between the two fields.

When an extension is Galois, we can give its group a special structure called the Krull topology. This turns it into what is known as a profinite group.