Editing Galois group

{{short description|Mathematical group}}
{{distinguish|Galois fields}}
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 (mathematics)|group]] associated with the field extension. The study of field extensions and their relationship to the [[polynomial]]s that give rise to them via Galois groups is called [[Galois theory]], so named in honor of [[Évariste Galois]] who first discovered them.

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

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

If <math>E/F</math> is a [[Galois extension]], then <math>\operatorname{Aut}(E/F)</math> is called the '''Galois group''' of <math>E/F</math>, and is usually denoted by <math>\operatorname{Gal}(E/F)</math>.<ref>Some authors refer to <math>\operatorname{Aut}(E/F)</math> as the Galois group for arbitrary extensions <math>E/F</math> and use the corresponding notation, e.g. {{harvnb|Jacobson|2009}}.</ref>

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

=== Galois group of a polynomial ===
Another definition of the Galois group comes from the Galois group of a polynomial <math>f \in F[x]</math>. If there is a field <math>K/F</math> such that <math>f</math> factors as a product of linear polynomials

:<math>f(x) = (x-\alpha_1)\cdots (x - \alpha_k) \in K[x]</math>

over the field <math>K</math>, then the '''Galois group of the polynomial''' <math>f</math> is defined as the Galois group of <math>K/F</math> where <math>K</math> 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 <math>K/k</math>, there is a bijection between the set of subfields <math>k \subset E \subset K</math> and the subgroups <math>H \subset G.</math> Then, <math>E</math> is given by the set of invariants of <math>K</math> under the action of <math>H</math>, so

:<math>E = K^H = \{ a\in K : ga = a \text{ where } g \in H \}</math>

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

=== Lattice structure ===
Suppose <math>K_1,K_2</math> are Galois extensions of <math>k</math> with Galois groups <math>G_1,G_2.</math> The field <math>K_1K_2</math> with Galois group <math>G = \operatorname{Gal}(K_1K_2/k)</math> has an injection <math>G \to G_1 \times G_2</math> which is an isomorphism whenever <math>K_1 \cap K_2 = k</math>.<ref name=":1" />

==== Inducting ====
As a corollary, this can be inducted finitely many times. Given Galois extensions <math>K_1,\ldots, K_n / k</math> where <math>K_{i+1} \cap (K_1\cdots K_i) = k,</math> then there is an isomorphism of the corresponding Galois groups:

:<math>\operatorname{Gal}(K_1\cdots K_n/k) \cong \operatorname{Gal}(K_1/k)\times \cdots \times \operatorname{Gal}(K_n/k).</math>

==Examples==
In the following examples <math>F</math> is a field, and <math>\Complex, \R, \Q</math> are the fields of [[complex number|complex]], [[real number|real]], and [[rational number|rational]] numbers, respectively. The notation {{math|''F''(''a'')}} indicates the field extension obtained by [[adjunction (field theory)|adjoining]] an element {{math|''a''}} to the field {{math|''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 group<ref name=":0">{{Cite web| url=http://abstract.ups.edu/download/aata-20100827.pdf |archive-url=https://web.archive.org/web/20111218100650/http://abstract.ups.edu/download/aata-20100827.pdf |archive-date=2011-12-18 |url-status=live |title=Abstract Algebra| pages=372–377}}</ref> of a finite field extension is the following: Given a polynomial <math>f(x) \in F[x]</math>, let <math>E/F</math> be its splitting field extension. Then the order of the Galois group is equal to the degree of the field extension; that is,

:<math>\left|\operatorname{Gal}(E/F)\right| = [E:F]</math>

==== Eisenstein's criterion ====
A useful tool for determining the Galois group of a polynomial comes from [[Eisenstein's criterion]]. If a polynomial <math>f \in F[x]</math> factors into irreducible polynomials <math>f = f_1\cdots f_k</math> the Galois group of <math>f</math> can be determined using the Galois groups of each <math>f_i</math> since the Galois group of <math>f</math> contains each of the Galois groups of the <math>f_i.</math>

=== Trivial group ===
<math>\operatorname{Gal}(F/F)</math> is the trivial group that has a single element, namely the identity automorphism.

Another example of a Galois group which is trivial is <math>\operatorname{Aut}(\R/\Q).</math> Indeed, it can be shown that any automorphism of <math>\R</math> must preserve the [[order theory|ordering]] of the real numbers and hence must be the identity.

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

:<math>\exp \left (\tfrac{2\pi i}{3} \right ) \sqrt[3]{2}</math> and <math>\exp \left (\tfrac{4\pi i}{3} \right ) \sqrt[3]{2},</math>

are missing from the extension—in other words {{math|''K''}} is not a [[splitting field]].

=== Finite abelian groups ===
The Galois group <math>\operatorname{Gal}(\Complex/\R)</math> has two elements, the identity automorphism and the [[complex conjugation]] automorphism.<ref>{{citation|title=Classical Algebra: Its Nature, Origins, and Uses|first=Roger L.|last=Cooke|publisher=John Wiley & Sons| year=2008| isbn=9780470277973|page=138|url=https://books.google.com/books?id=JG-skeT1eWAC&pg=PA138}}.</ref>

==== Quadratic extensions ====
The degree two field extension <math>\Q(\sqrt{2})/\Q</math> has the Galois group <math>\operatorname{Gal}(\Q(\sqrt{2})/\Q)</math> with two elements, the identity automorphism and the automorphism <math>\sigma</math> which exchanges <math>\sqrt2</math> and <math>-\sqrt2</math>. This example generalizes for a prime number <math>p \in \N.</math>

==== Product of quadratic extensions ====
Using the lattice structure of Galois groups, for non-equal prime numbers <math>p_1, \ldots, p_k</math> the Galois group of <math>\Q \left (\sqrt{p_1},\ldots, \sqrt{p_k} \right)/\Q</math> is

:<math>\operatorname{Gal} \left (\Q(\sqrt{p_1},\ldots, \sqrt{p_k})/\Q \right ) \cong \operatorname{Gal}\left (\Q(\sqrt{p_1})/\Q \right )\times \cdots \times \operatorname{Gal} \left (\Q(\sqrt{p_k})/\Q \right ) \cong (\Z/2\Z)^k</math>

==== Cyclotomic extensions ====
Another useful class of examples comes from the splitting fields of [[cyclotomic polynomial]]s. These are polynomials <math>\Phi_n</math> defined as

:<math>\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)</math>

whose degree is <math>\phi(n)</math>, [[Euler's totient function]] at <math>n</math>. Then, the splitting field over <math>\Q</math> is <math>\Q(\zeta_n)</math> and has automorphisms <math>\sigma_a</math> sending <math>\zeta_n \mapsto \zeta_n^a</math> for <math>1 \leq a < n</math> relatively prime to <math>n</math>. Since the degree of the field is equal to the degree of the polynomial, these automorphisms generate the Galois group.<ref>{{Cite book| last1=Dummit| title=Abstract Algebra| last2=Foote |pages=596, 14.5 Cyclotomic Extensions}}</ref> If <math>n = p_1^{a_1}\cdots p_k^{a_k},</math> then

:<math>\operatorname{Gal}(\Q(\zeta_n)/\Q) \cong \prod_{a_i} \operatorname{Gal}\left (\Q(\zeta_{p_i^{a_i}})/\Q \right )</math>

If <math>n</math> is a prime <math>p </math>, then a corollary of this is

:<math>\operatorname{Gal}(\Q(\zeta_p)/\Q) \cong \Z/(p-1)\Z</math>

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 {{math|''q''}} is a prime power, and if <math>F = \mathbb{F}_q</math> and <math>E=\mathbb{F}_{q^n}</math> denote the [[Finite field|Galois fields]] of order <math>q</math> and <math>q^n</math> respectively, then <math>\operatorname{Gal}(E/F)</math> is cyclic of order {{math|''n''}} and generated by the [[Frobenius homomorphism]].

==== Degree 4 examples ====
The field extension <math>\Q(\sqrt{2},\sqrt{3})/\Q</math> is an example of a degree <math>4</math> field extension.<ref>Since <math>\Q(\sqrt{2},\sqrt{3}) = \Q\oplus \Q\cdot\sqrt{2} \oplus \Q\cdot\sqrt{3} \oplus \Q\cdot \sqrt{6}</math> as a <math>\Q</math> vector space.</ref> This has two automorphisms <math>\sigma, \tau</math> where <math>\sigma(\sqrt{2}) = -\sqrt{2}</math> and <math>\tau(\sqrt{3})=-\sqrt{3}.</math> Since these two generators define a group of order <math>4</math>, the [[Klein four-group]], they determine the entire Galois group.<ref name=":0" />

Another example is given from the splitting field <math>E/\Q</math> of the polynomial

:<math>f(x) = x^4 + x^3 + x^2 + x + 1</math>

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

:<math>\begin{cases}\sigma_l : E \to E \\ \sigma_2 : \exp \left (\frac{2\pi i}{5} \right) \mapsto \left (\exp \left (\frac{2\pi i}{5} \right ) \right )^l \end{cases}</math>

generating a group of order <math>4</math>. Since <math>\sigma_2</math> generates this group, the Galois group is isomorphic to <math>\Z/4\Z</math>.

=== Finite non-abelian groups ===
Consider now <math>L = \Q(\sqrt[3]{2}, \omega),</math> where <math>\omega</math> is a [[root of unity|primitive cube root of unity]]. The group <math>\operatorname{Gal}(L/\Q)</math> is isomorphic to {{math|''S''<sub>3</sub>}}, the [[dihedral group of order 6]], and {{math|''L''}} is in fact the splitting field of <math>x^3-2</math> over <math>\Q.</math>

==== Quaternion group ====
The [[Quaternion group]] can be found as the Galois group of a field extension of <math>\Q</math>. For example, the field extension

:<math>\Q \left (\sqrt{2}, \sqrt{3}, \sqrt{(2+\sqrt{2})(3+\sqrt{3})} \right )</math>

has the prescribed Galois group.<ref>{{Cite book| last=Milne| url= https://www.jmilne.org/math/CourseNotes/ft.html| title=Field Theory|pages=46}}</ref>

==== Symmetric group of prime order ====
If <math>f</math> is an [[irreducible polynomial]] of prime degree <math>p</math> with rational coefficients and exactly two non-real roots, then the Galois group of <math>f</math> is the full [[symmetric group]] <math>S_p.</math><ref name=":1">{{Cite book| last=Lang| first=Serge| title=Algebra| edition=Revised Third| pages=263, 273}}</ref>

For example, <math>f(x)=x^5-4x+2 \in \Q[x]</math> is irreducible from Eisenstein's criterion. Plotting the graph of <math>f</math> with graphing software or paper shows it has three real roots, hence two complex roots, showing its Galois group is <math>S_5</math>.

=== Comparing Galois groups of field extensions of global fields ===
Given a [[global field]] extension <math>K/k</math> (such as <math>\mathbb{Q}(\sqrt[5]{3},\zeta_5 )/\mathbb{Q}</math>) and equivalence classes of valuations  <math>w</math> on <math>K</math> (such as the <math>p</math>-adic [[Valuation (algebra)|valuation]]) and <math>v</math> on <math>k</math> such that their completions give a Galois field extension<blockquote><math>K_w/k_v</math></blockquote>of [[local field]]s, there is an induced action of the Galois group <math>G = \operatorname{Gal}(K/k)</math> on the set of equivalence classes of valuations such that the completions of the fields are compatible. This means if <math>s \in G</math> then there is an induced isomorphism of local fields<blockquote><math>s_w:K_w \to K_{sw}</math></blockquote>Since we have taken the hypothesis that <math>w</math> lies over <math>v</math> (i.e. there is a Galois field extension <math>K_w/k_v</math>), the field morphism <math>s_w</math> is in fact an isomorphism of <math>k_v</math>-algebras. If we take the isotropy subgroup of <math>G</math> for the valuation class <math>w</math><blockquote><math>G_w = \{s \in G : sw = w \}</math></blockquote>then there is a surjection of the global Galois group to the local Galois group such that there is an isomorphism between the local Galois group and the isotropy subgroup. Diagrammatically, this means<blockquote><math>\begin{matrix}
\operatorname{Gal}(K/v)& \twoheadrightarrow & \operatorname{Gal}(K_w/k_v) \\
\downarrow & & \downarrow \\
G & \twoheadrightarrow & G_w
\end{matrix}</math></blockquote>where the vertical arrows are isomorphisms.<ref>{{Cite web| title=Comparing the global and local galois groups of an extension of number fields| url=https://math.stackexchange.com/q/2470699 | access-date=2020-11-11|website=Mathematics Stack Exchange}}</ref> 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 <math>\operatorname{Aut}(\Complex/\Q)</math>, since it contains every algebraic field extension <math>E/\Q</math>. For example, the field extensions <math>\Q(\sqrt{a})/\Q</math> for a square-free element <math>a \in \Q</math> each have a unique degree <math>2</math> automorphism, inducing an automorphism in <math>\operatorname{Aut}(\Complex/\Q).</math>

One of the most studied classes of infinite Galois group is the [[absolute Galois group]], which is an infinite, [[profinite group|profinite]] group defined as the [[inverse limit]] of all finite Galois extensions <math>E/F</math> for a fixed field. The inverse limit is denoted

:<math>\operatorname{Gal}(\overline{F}/F) := \varprojlim_{E/F \text{ finite separable}}{\operatorname{Gal}(E/F)}</math>,

where <math>\overline{F}</math> is the separable closure of the field <math>F</math>. Note this group is a [[topological group]].<ref>{{Cite web| url=https://stacks.math.columbia.edu/tag/0BMI|title=9.22 Infinite Galois theory|website=The Stacks project}}</ref> Some basic examples include <math>\operatorname{Gal}(\overline{\Q}/\Q)</math> and

:<math>\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q) \cong \hat{\Z} \cong \prod_p \Z_p</math>.<ref>{{Cite web| url=https://www.jmilne.org/math/CourseNotes/FT.pdf |archive-url=https://web.archive.org/web/20080827191224/http://jmilne.org/math/CourseNotes/FT.pdf |archive-date=2008-08-27 |url-status=live |title=Field Theory|last=Milne|page=98}}</ref><ref>{{Cite web| url=http://diposit.ub.edu/dspace/bitstream/2445/122676/2/memoria.pdf| title= Infinite Galois Theory |pages=14|url-status=live|archive-url= https://web.archive.org/web/20200406024608/http://diposit.ub.edu/dspace/bitstream/2445/122676/2/memoria.pdf|archive-date=6 April 2020}}</ref>

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

:<math>\operatorname{Gal}(\Q(\sqrt{2},\sqrt{3},\sqrt{5}, \ldots)/ \Q) \cong \prod_{p} \Z/2</math>,

which can be deduced from the profinite limit

:<math>\cdots \to \operatorname{Gal}(\Q(\sqrt{2},\sqrt{3},\sqrt{5})/\Q) \to \operatorname{Gal}(\Q(\sqrt{2},\sqrt{3})/\Q) \to \operatorname{Gal}(\Q(\sqrt{2})/\Q)</math>

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 <math>E/F</math> is a Galois extension, then <math>\operatorname{Gal}(E/F)</math> can be given a [[topological space|topology]], called the Krull topology, that makes it into a [[profinite group]].

==See also==
*[[Fundamental theorem of Galois theory]]
*[[Absolute Galois group]]
*[[Galois representation]]
*[[Demushkin group]]
*[[Solvable group]]

==Notes==
{{reflist}}

==References==
*{{cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| title=Basic Algebra I| year=2009| edition=2nd| publisher=Dover Publications| isbn=978-0-486-47189-1| orig-year=1985}}
*{{Lang Algebra}}
* [https://www.math.colostate.edu/~hulpke/paper/gov.pdf Hulpke, Alexander (1999). "Techniques for the Computation of Galois Groups". In: Matzat, B.H., Greuel, GM., Hiss, G. (eds) ''Algorithmic Algebra and Number Theory''. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59932-3_4]

==External links==
*{{springer|title=Galois group|id=p/g043150}}
*[https://math.stackexchange.com/q/248224 Galois group and the Quaternion group]
*{{MathPages|id=home/kmath290/kmath290|title=Galois Groups}}
*[https://math.stackexchange.com/q/2470699 Comparing the global and local galois groups of an extension of number fields]
*[https://www.math.ias.edu/~rtaylor/longicm02.pdf Galois Representations] - [[Richard Taylor (mathematician)|Richard Taylor]]

[[Category:Galois theory]]