Galois group

This article defines a natural context where a group occurs, or is associated, with another algebraic, topological, analytic or discrete structure
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This term is related to: Galois theory
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This article gives a basic definition in the following area: Galois theory
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Definition

Symbol-free definition

Let $L/K$ be an extension of fields that is a Galois extension (that is, it is algebraic, normal and separable).

The Galois group of this extension is defined as the group of field automorphisms of $L$ which fix every element inside $K$.

The Galois group is often then written $\mathrm {Gal} (L/K)$.

Examples

• The Galois group of the trivial field extension $K/K$ of any field $K$ is the trivial group.
• The Galois group of $L=\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})$ is the group of field automorphisms of $L$ fixing $\mathbb {Q}$. Furthermore, it can be shown that such an automorphism is determined by its effects on ${\sqrt {2}}$ and ${\sqrt {3}}$. Since this is an extension of degree 4, the order of the Galois group is 4. Consider the element $\sigma \in \mathrm {Gal} (L/\mathbb {Q} )$ such that $\sigma ({\sqrt {2}})=-{\sqrt {2}}$ and $\sigma ({\sqrt {3}})={\sqrt {3}}$, as well as the element $\tau \in \mathrm {Gal} (L/\mathbb {Q} )$ such that $\tau ({\sqrt {2}})={\sqrt {2}}$ and $\tau ({\sqrt {3}})=-{\sqrt {3}}$. We can check that $\tau \sigma =\sigma \tau$ and that $\tau ^{2}=\sigma ^{2}=\mathrm {id}$. This determines that $\mathrm {Gal} (L/\mathbb {Q} )$ is isomorphic to the Klein four-group.

• More generally, by similar logic, if $p_{1},\dots ,p_{n}$ are distinct primes, $\mathbb {Q} ({\sqrt {p_{1}}},\dots ,{\sqrt {p_{n}}})$ is the direct product of $n$ copies of cyclic group:Z2, that is, it is the elementary abelian group of order $2^{n}$.

Facts

Realization

The question: does every finite group occur as the Galois group of some Galois extension is the famous inverse Galois problem. It has been proved that every solvable group occurs as the Galois group of some group.

Galois group of a polynomial

Further information: Galois group of a polynomial

While this page deals with the definition of a Galois group of a field extension, we can also talk about the Galois group of a polynomial. For $K$ a field, the Galois group of a polynomial $f(X)\in K[X]$ is defined to be the Galois group of a splitting field for $f(X)$ over $K$.

Representations of the Galois group

A permutation representation

Consider a Galois extension of finite degree. Then, by the primitive element theorem, there exists a primitive element for the extension, or in other words, there exists a monic polynomial over $K$ such that the extension is generated by any root of $K$.

Now, every element of the Galois group acts as a permutation on the roots of this polynomial. In fact, the action is transitive (since the polynomial is irreducible) on the roots. This thus gives a transitive permutation representation of the Galois group.

Note, however, that the precise transitive permutation representation depends on the choice of the irreducible polynomial. Or does it?

A linear representation

Since the Galois group gives linear automorphisms of $L$ over $K$, it naturally gives rise to a linear representation.

Galois cohomology

Another rich source of representations (of all kinds) of the Galois group, is Galois cohomology.