# Galois group

Last updated

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

## Contents

For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.

## Definition

Suppose that ${\displaystyle E}$ is an extension of the field ${\displaystyle F}$ (written as ${\displaystyle E/F}$ and read "E over F"). An automorphism of ${\displaystyle E/F}$ is defined to be an automorphism of ${\displaystyle E}$ that fixes ${\displaystyle F}$ pointwise. In other words, an automorphism of ${\displaystyle E/F}$ is an isomorphism ${\displaystyle \alpha :E\to E}$ such that ${\displaystyle \alpha (x)=x}$ for each ${\displaystyle x\in F}$. The set of all automorphisms of ${\displaystyle E/F}$ forms a group with the operation of function composition. This group is sometimes denoted by ${\displaystyle \operatorname {Aut} (E/F).}$

If ${\displaystyle E/F}$ is a Galois extension, then ${\displaystyle \operatorname {Aut} (E/F)}$ is called the Galois group of ${\displaystyle E/F}$, and is usually denoted by ${\displaystyle \operatorname {Gal} (E/F)}$. [1]

If ${\displaystyle E/F}$ is not a Galois extension, then the Galois group of ${\displaystyle E/F}$ is sometimes defined as ${\displaystyle \operatorname {Aut} (K/F)}$, where ${\displaystyle K}$ is the Galois closure of ${\displaystyle E}$.

### Galois group of a polynomial

Another definition of the Galois group comes from the Galois group of a polynomial ${\displaystyle f\in F[x]}$. If there is a field ${\displaystyle K/F}$ such that ${\displaystyle f}$ factors as a product of linear polynomials

${\displaystyle f(x)=(x-\alpha _{1})\cdots (x-\alpha _{k})\in K[x]}$

over the field ${\displaystyle K}$, then the Galois group of the polynomial${\displaystyle f}$ is defined as the Galois group of ${\displaystyle K/F}$ where ${\displaystyle K}$ is minimal among all such fields.

## Structure of Galois groups

### Fundamental theorem of Galois theory

One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension ${\displaystyle K/k}$, there is a bijection between the set of subfields ${\displaystyle k\subset E\subset K}$ and the subgroups ${\displaystyle H\subset G.}$ Then, ${\displaystyle E}$ is given by the set of invariants of ${\displaystyle K}$ under the action of ${\displaystyle H}$, so

${\displaystyle E=K^{H}=\{a\in K:ga=a{\text{ where }}g\in H\}}$

Moreover, if ${\displaystyle H}$ is a normal subgroup then ${\displaystyle G/H\cong \operatorname {Gal} (E/k)}$. And conversely, if ${\displaystyle E/k}$ is a normal field extension, then the associated subgroup in ${\displaystyle \operatorname {Gal} (K/k)}$ is a normal group.

### Lattice structure

Suppose ${\displaystyle K_{1},K_{2}}$ are Galois extensions of ${\displaystyle k}$ with Galois groups ${\displaystyle G_{1},G_{2}.}$ The field ${\displaystyle K_{1}K_{2}}$ with Galois group ${\displaystyle G=\operatorname {Gal} (K_{1}K_{2}/k)}$ has an injection ${\displaystyle G\to G_{1}\times G_{2}}$ which is an isomorphism whenever ${\displaystyle K_{1}\cap K_{2}=k}$. [2]

#### Inducting

As a corollary, this can be inducted finitely many times. Given Galois extensions ${\displaystyle K_{1},\ldots ,K_{n}/k}$ where ${\displaystyle K_{i+1}\cap (K_{1}\cdots K_{i})=k,}$ then there is an isomorphism of the corresponding Galois groups:

${\displaystyle \operatorname {Gal} (K_{1}\cdots K_{n}/k)\cong G_{1}\times \cdots \times G_{n}.}$

## Examples

In the following examples ${\displaystyle F}$ is a field, and ${\displaystyle \mathbb {C} ,\mathbb {R} ,\mathbb {Q} }$ are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension obtained by adjoining an element a to the field F.

### Computational tools

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

One of the basic propositions required for completely determining the Galois groups [3] of a finite field extension is the following: Given a polynomial ${\displaystyle f(x)\in F[x]}$, let ${\displaystyle E/F}$ be its splitting field extension. Then the order of the Galois group is equal to the degree of the field extension; that is,

${\displaystyle |\operatorname {Gal} (E/F)|=[E:F]}$

#### Eisenstein's criterion

A useful tool for determining the Galois group of a polynomial comes from Eisenstein's criterion. If a polynomial ${\displaystyle f\in F[x]}$ factors into irreducible polynomials ${\displaystyle f=f_{1}\cdots f_{k}}$ the Galois group of ${\displaystyle f}$ can be determined using the Galois groups of each ${\displaystyle f_{i}}$ since the Galois group of ${\displaystyle f}$ contains each of the Galois groups of the ${\displaystyle f_{i}.}$

### Trivial group

${\displaystyle \operatorname {Gal} (F/F)}$ is the trivial group that has a single element, namely the identity automorphism.

Another example of a Galois group which is trivial is ${\displaystyle \operatorname {Aut} (\mathbb {R} /\mathbb {Q} ).}$ Indeed, it can be shown that any automorphism of ${\displaystyle \mathbb {R} }$ must preserve the ordering of the real numbers and hence must be the identity.

Consider the field ${\displaystyle K=\mathbb {Q} ({\sqrt[{3}]{2}}).}$ The group ${\displaystyle \operatorname {Aut} (K/\mathbb {Q} )}$ contains only the identity automorphism. This is because ${\displaystyle K}$ is not a normal extension, since the other two cube roots of ${\displaystyle 2}$,

${\displaystyle \exp \left({\tfrac {2\pi i}{3}}\right){\sqrt[{3}]{2}}}$ and ${\displaystyle \exp \left({\tfrac {4\pi i}{3}}\right){\sqrt[{3}]{2}},}$

are missing from the extension—in other words K is not a splitting field.

### Finite abelian groups

The Galois group ${\displaystyle \operatorname {Gal} (\mathbb {C} /\mathbb {R} )}$ has two elements, the identity automorphism and the complex conjugation automorphism. [4]

The degree two field extension ${\displaystyle \mathbb {Q} ({\sqrt {2}})/\mathbb {Q} }$ has the Galois group ${\displaystyle \operatorname {Gal} (\mathbb {Q} ({\sqrt {2}})/\mathbb {Q} ).}$ with two elements, the identity automorphism and the automorphism ${\displaystyle \sigma }$ which exchanges 2 and −2. This example generalizes for a prime number ${\displaystyle p\in \mathbb {N} .}$

Using the lattice structure of Galois groups, for non-equal prime numbers ${\displaystyle p_{1},\ldots ,p_{k}}$ the Galois group of ${\displaystyle \mathbb {Q} \left({\sqrt {p_{1}}},\ldots ,{\sqrt {p_{k}}}\right)/\mathbb {Q} }$ is

${\displaystyle \operatorname {Gal} \left(\mathbb {Q} ({\sqrt {p_{1}}},\ldots ,{\sqrt {p_{k}}})/\mathbb {Q} \right)\cong \operatorname {Gal} \left(\mathbb {Q} ({\sqrt {p_{1}}})/\mathbb {Q} \right)\times \cdots \times \operatorname {Gal} \left(\mathbb {Q} ({\sqrt {p_{k}}})/\mathbb {Q} \right)\cong \mathbb {Z} /2\times \cdots \times \mathbb {Z} /2}$

#### Cyclotomic extensions

Another useful class of examples comes from the splitting fields of cyclotomic polynomials. These are polynomials ${\displaystyle \Phi _{n}}$ defined as

${\displaystyle \Phi _{n}(x)=\prod _{\begin{matrix}1\leq k\leq n\\\gcd(k,n)=1\end{matrix}}\left(x-e^{\frac {2ik\pi }{n}}\right)}$

whose degree is ${\displaystyle \phi (n)}$, Euler's totient function at ${\displaystyle n}$. Then, the splitting field over ${\displaystyle \mathbb {Q} }$ is ${\displaystyle \mathbb {Q} (\zeta _{n})}$ and has automorphisms ${\displaystyle \sigma _{a}}$ sending ${\displaystyle \zeta _{n}\mapsto \zeta _{n}^{a}}$ for ${\displaystyle 1\leq a relatively prime to ${\displaystyle n}$. Since the degree of the field is equal to the degree of the polynomial, these automorphisms generate the Galois group. [5] If ${\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}},}$ then

${\displaystyle \operatorname {Gal} (\mathbb {Q} (\zeta _{n})/\mathbb {Q} )\cong \prod _{a_{i}}\operatorname {Gal} \left(\mathbb {Q} (\zeta _{p_{i}^{a_{i}}})/\mathbb {Q} \right)}$

If ${\displaystyle n}$ is a prime ${\displaystyle p}$, then a corollary of this is

${\displaystyle \operatorname {Gal} (\mathbb {Q} (\zeta _{p})/\mathbb {Q} )\cong \mathbb {Z} /(p-1)}$

In fact, any finite abelian group can be found as the Galois group of some subfield of a cyclotomic field extension by the Kronecker–Weber theorem.

#### Finite fields

Another useful class of examples of Galois groups with finite abelian groups comes from finite fields. If q is a prime power, and if ${\displaystyle F=\mathbb {F} _{q}}$ and ${\displaystyle E=\mathbb {F} _{q^{n}}}$ denote the Galois fields of order ${\displaystyle q}$ and ${\displaystyle q^{n}}$ respectively, then ${\displaystyle \operatorname {Gal} (E/F)}$ is cyclic of order n and generated by the Frobenius homomorphism.

#### Degree 4 examples

The field extension ${\displaystyle \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} }$ is an example of a degree ${\displaystyle 4}$ field extension. [6] This has two automorphisms ${\displaystyle \sigma ,\tau }$ where ${\displaystyle \sigma ({\sqrt {2}})=-{\sqrt {2}}}$ and ${\displaystyle \tau ({\sqrt {3}})=-{\sqrt {3}}.}$ Since these two generators define a group of order ${\displaystyle 4}$, the Klein four-group, they determine the entire Galois group. [3]

Another example is given from the splitting field ${\displaystyle E/\mathbb {Q} }$ of the polynomial

${\displaystyle f(x)=x^{4}+x^{3}+x^{2}+x+1}$

Note because ${\displaystyle (x-1)f(x)=x^{5}-1,}$ the roots of ${\displaystyle f(x)}$ are ${\displaystyle \exp \left({\tfrac {2k\pi i}{5}}\right).}$ There are automorphisms

${\displaystyle {\begin{cases}\sigma _{l}:E\to E\\\exp \left({\frac {2\pi i}{5}}\right)\mapsto \left(\exp \left({\frac {2\pi i}{5}}\right)\right)^{l}\end{cases}}}$

generating a group of order ${\displaystyle 4}$. Since ${\displaystyle \sigma _{1}}$ generates this group, the Galois group is isomorphic to ${\displaystyle \mathbb {Z} /4}$.

### Finite non-abelian groups

Consider now ${\displaystyle L=\mathbb {Q} ({\sqrt[{3}]{2}},\omega ),}$ where ${\displaystyle \omega }$ is a primitive cube root of unity. The group ${\displaystyle \operatorname {Gal} (L/\mathbb {Q} )}$ is isomorphic to S3, the dihedral group of order 6, and L is in fact the splitting field of ${\displaystyle x^{3}-2}$ over ${\displaystyle \mathbb {Q} .}$

#### Quaternion group

The Quaternion group can be found as the Galois group of a field extension of ${\displaystyle \mathbb {Q} }$. For example, the field extension

${\displaystyle \mathbb {Q} \left({\sqrt {2}},{\sqrt {3}},{\sqrt {(2+{\sqrt {2}})(3+{\sqrt {3}})}}\right)}$

has the prescribed Galois group. [7]

#### Symmetric group of prime order

If ${\displaystyle f}$ is an irreducible polynomial of prime degree ${\displaystyle p}$ with rational coefficients and exactly two non-real roots, then the Galois group of ${\displaystyle f}$ is the full symmetric group ${\displaystyle S_{p}.}$ [2]

For example, ${\displaystyle f(x)=x^{5}-4x+2\in \mathbb {Q} [x]}$ is irreducible from Eisenstein's criterion. Plotting the graph of ${\displaystyle f}$ with graphing software or paper shows it has three real roots, hence two complex roots, showing its Galois group is ${\displaystyle S_{5}}$.

### Comparing Galois groups of field extensions of global fields

Given a global field extension ${\displaystyle K/k}$ (such as ${\displaystyle \mathbb {Q} ({\sqrt[{5}]{3}},\zeta _{5})/\mathbb {Q} }$) the and ${\displaystyle w}$ an equivalence class of valuations on ${\displaystyle K}$ (such as the ${\displaystyle p}$-adic valuation), and ${\displaystyle v}$ on ${\displaystyle k}$ such that their completions give a Galois field extension

${\displaystyle K_{w}/k_{v}}$

of local fields. Then, there is an induced action of the Galois group

${\displaystyle G=\operatorname {Gal} (K/k)}$

on the set of equivalence classes of valuations such that the completions of the fields are compatible. This means if ${\displaystyle s\in G}$ then there is an induced isomorphic of local fields

${\displaystyle s_{w}:K_{w}\to K_{sw}}$

Since we have taken the hypothesis that ${\displaystyle w}$ lies over ${\displaystyle v}$ (i.e. there is a Galois field extension ${\displaystyle K_{w}/k_{v}}$), the field morphism ${\displaystyle s_{w}}$ is in fact an isomorphism of ${\displaystyle k_{v}}$-algebras. If we take the isotropy subgroup of ${\displaystyle G}$ for the valuation class ${\displaystyle w}$

${\displaystyle G_{w}=\{s\in G:sw=w\}}$

then there's a surjection of the global Galois group to the local Galois group such that there's and isomorphism between the local Galois group and the isotropy subgroup. Diagrammatically, this means

${\displaystyle {\begin{matrix}\operatorname {Gal} (K/v)&\twoheadrightarrow &\operatorname {Gal} (K_{w}/k_{v})\\\downarrow &&\downarrow \\G&\twoheadrightarrow &G_{w}\end{matrix}}}$

where the vertical arrows are isomorphisms. [8] This gives a technique for constructing Galois groups of local fields using global Galois groups.

### Infinite groups

A basic example of a field extension with an infinite group of automorphisms, is ${\displaystyle \operatorname {Aut} (\mathbb {C} /\mathbb {Q} )}$ since it contains every algebraic field extension ${\displaystyle E/\mathbb {Q} }$. For example, the field extensions ${\displaystyle \mathbb {Q} ({\sqrt {a}})/\mathbb {Q} }$ for a square-free element ${\displaystyle a\in \mathbb {Q} }$ each have a unique degree ${\displaystyle 2}$ automorphism, inducing an automorphism in ${\displaystyle \operatorname {Aut} (\mathbb {C} /\mathbb {Q} ).}$

One of the most studied classes of examples of infinite Galois groups come from the Absolute Galois group, which are profinite groups. These are infinite groups defined as the inverse limit of Galois groups all finite Galois extensions ${\displaystyle E/F}$ for a fixed field. The inverse limit is denoted

${\displaystyle \operatorname {Gal} ({\overline {F}}/F):=\varprojlim _{E/F{\text{ finite separable}}}{\operatorname {Gal} (E/F)}}$

where ${\displaystyle {\overline {F}}}$ is the separable closure of a field. Note this group is a Topological group. [9] Some basic examples include ${\displaystyle \operatorname {Gal} ({\overline {\mathbb {Q} }}/\mathbb {Q} )}$ and

${\displaystyle \operatorname {Gal} ({\overline {\mathbb {F} }}_{q}/\mathbb {F} _{q})\cong {\hat {\mathbb {Z} }}\cong \prod _{p}\mathbb {Z} _{p}}$ [10] [11]

Another readily computable example comes from the field extension ${\displaystyle \mathbb {Q} ({\sqrt {2}},{\sqrt {3}},{\sqrt {5}},\ldots )/\mathbb {Q} }$ containing the square root of every positive prime. It has Galois group

${\displaystyle \operatorname {Gal} (\mathbb {Q} ({\sqrt {2}},{\sqrt {3}},{\sqrt {5}},\ldots )/\mathbb {Q} )\cong \prod _{p}\mathbb {Z} /2}$

which can be deduced from the profinite limit

${\displaystyle \cdots \to \operatorname {Gal} (\mathbb {Q} ({\sqrt {2}},{\sqrt {3}},{\sqrt {5}})/\mathbb {Q} )\to \operatorname {Gal} (\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} )\to \operatorname {Gal} (\mathbb {Q} ({\sqrt {2}})/\mathbb {Q} )}$

and using the computation of the Galois groups.

## Properties

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of the Galois group correspond to the intermediate fields of the field extension.

If ${\displaystyle E/F}$ is a Galois extension, then ${\displaystyle \operatorname {Gal} (E/F)}$ can be given a topology, called the Krull topology, that makes it into a profinite group.

## Notes

1. Some authors refer to ${\displaystyle \operatorname {Aut} (E/F)}$ as the Galois group for arbitrary extensions ${\displaystyle E/F}$ and use the corresponding notation, e.g. Jacobson 2009.
2. Lang, Serge. Algebra (Revised Third ed.). pp. 263, 273.
3. "Abstract Algebra" (PDF). pp. 372–377.
4. Cooke, Roger L. (2008), Classical Algebra: Its Nature, Origins, and Uses, John Wiley & Sons, p. 138, ISBN   9780470277973 .
5. Dummit; Foote. Abstract Algebra. pp. 596, 14.5 Cyclotomic Extensions.
6. Since ${\displaystyle \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})=\mathbb {Q} \oplus \mathbb {Q} \cdot {\sqrt {2}}\oplus \mathbb {Q} \cdot {\sqrt {3}}\oplus \mathbb {Q} \cdot {\sqrt {6}}}$ as a ${\displaystyle \mathbb {Q} }$ vector space.
7. Milne. Field Theory. p. 46.
8. "Comparing the global and local galois groups of an extension of number fields". Mathematics Stack Exchange. Retrieved 2020-11-11.
9. "9.22 Infinite Galois theory". The Stacks project.
10. Milne. "Field Theory" (PDF). p. 98.
11. "Infinite Galois Theory" (PDF). p. 14. Archived (PDF) from the original on 6 April 2020.

## Related Research Articles

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, a finite field or Galois field is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number.

In mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of E are those of F restricted to E. In this case, F is an extension field of E and E is a subfield of F. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.

In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.

In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer n – and therefore belong to abstract algebra. The theory of cyclic extensions of the field K when the characteristic of K does divide n is called Artin–Schreier theory.

In abstract algebra, a normal extension is an algebraic field extension L/K for which every polynomial that is irreducible over K either has no root in L or splits into linear factors in L. Bourbaki calls such an extension a quasi-Galois extension.

In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory.

In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its p-th power. In certain contexts it is an automorphism, but this is not true in general.

In abstract algebra, Hilbert's Theorem 90 is an important result on cyclic extensions of fields that leads to Kummer theory. In its most basic form, it states that if L/K is a cyclic extension of fields with Galois group G = Gal(L/K) generated by an element and if is an element of L of relative norm 1, then there exists in L such that

In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of G need be considered, rather than two. This was certainly familiar before Hilbert.

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.

The Artin reciprocity law, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

In commutative algebra, an element b of a commutative ring B is said to be integral overA, a subring of B, if there are n ≥ 1 and aj in A such that

In field theory, a branch of mathematics, the minimal polynomial of a value α is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1, and the specified type for the remaining coefficients could be integers, rational numbers, real numbers, or others.

In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field, conjectured in 1993 and proven in 2003 by Kai Behrend. Unlike the classical one, the formula counts points in the "stacky way"; it takes into account the presence of nontrivial automorphisms.

In mathematics, a profinite integer is an element of the ring

In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the general linear group of X, the group of invertible linear transformations from X to itself.