Editing Inverse Galois problem

{{Short description|Unsolved problem in mathematics}}
{{unsolved|mathematics|Is every [[finite group]] the [[Galois group]] of a [[Galois extension]] of the [[rational number]]s?}}
In [[Galois theory]], the '''inverse Galois problem''' concerns whether or not every [[finite group]] appears as the [[Galois group]] of some [[Galois extension]] of the [[rational number]]s <math>\mathbb{Q}</math>. This problem, first posed in the early 19th century,<ref>{{Cite web |title=Mathematical Sciences Research Institute Publications 45 |url=http://library.msri.org/books/Book45/files/book45.pdf |website=MSRI |access-date=2016-04-17 |archive-date=2017-08-29 |archive-url=https://web.archive.org/web/20170829003635/http://library.msri.org/books/Book45/files/book45.pdf |url-status=dead }}</ref> is unsolved.

There are some [[permutation group]]s for which [[generic polynomial]]s are known, which define all [[algebraic extension]]s of <math>\mathbb{Q}</math> having a particular [[group (mathematics)|group]] as Galois group. These groups include all of degree no greater than {{math|5}}. There also are groups known not to have generic polynomials, such as the cyclic group of [[order (group theory)|order]] {{math|8}}.

More generally, let {{mvar|G}} be a given finite group, and {{mvar|K}} a field. If there is a Galois extension field {{math|''L''/''K''}} whose Galois group is [[group isomorphism|isomorphic]] to {{mvar|G}}, one says that '''{{mvar|G}} is realizable over {{mvar|K}}'''.

==Partial results==
Many cases are known. It is known that every finite group is realizable over any [[Function field of an algebraic variety|function field]] in one variable over the [[complex number]]s <math>\mathbb{C}</math>, and more generally over function fields in one variable over any [[algebraically closed field]] of [[characteristic (algebra)|characteristic]] zero. [[Igor Shafarevich]] showed that every finite [[solvable group]] is realizable over <math>\mathbb{Q}</math>.<ref>Igor R. Shafarevich, ''The imbedding problem for splitting extensions'', Dokl. Akad. Nauk SSSR '''120''' (1958), 1217-1219.</ref> It is also known that every simple [[sporadic group]], except possibly the [[Mathieu group]] {{math|''M''<sub>23</sub>}}, is realizable over <math>\mathbb{Q}</math>.<ref>p. 5 of Jensen et al., 2002</ref>

[[David Hilbert]] showed that this question is related to a [[rationality question]] for {{mvar|G}}:

:If {{mvar|K}} is any extension of <math>\mathbb{Q}</math> on which {{mvar|G}} acts as an [[automorphism group]], and the [[Invariant theory|invariant field]] {{math|''K<sup>G</sup>''}} is rational over {{nowrap|<math>\mathbb{Q}</math>,}} then {{mvar|G}} is realizable over {{nowrap|<math>\mathbb{Q}</math>.}}

Here ''rational'' means that it is a [[purely transcendental]] extension of <math>\mathbb{Q}</math>, generated by an [[algebraically independent]] set. This criterion can for example be used to show that all the [[symmetric group]]s are realizable.

Much detailed work has been carried out on the question, which is in no sense solved in general. Some of this is based on constructing {{mvar|G}} geometrically as a [[Galois covering]] of the [[projective line]]: in algebraic terms, starting with an extension of the field <math>\mathbb{Q}(t)</math> of [[rational function]]s in an indeterminate {{mvar|t}}. After that, one applies [[Hilbert's irreducibility theorem]] to specialise {{mvar|t}}, in such a way as to preserve the Galois group.

All permutation groups of degree 23 or less, except the [[Mathieu group]] {{math|''M''<sub>23</sub>}}, are known to be realizable over {{nowrap|<math>\mathbb{Q}</math>}}.<ref>{{Cite web|url=http://galoisdb.math.upb.de/|title=Home|website=galoisdb.math.upb.de}}</ref> <ref>{{Cite web|url=https://arxiv.org/abs/2411.07857|title=17T7 is a Galois group over the rationals}}</ref>

All 13 non-[[abelian group|abelian]] [[simple group]]s smaller than PSL(2,25) (order 7800) are known to be realizable over {{nowrap|<math>\mathbb{Q}</math>.}}<ref>Malle and Matzat (1999), pp. 403-424</ref>

==A simple example: cyclic groups==
It is possible, using classical results, to  construct explicitly a [[polynomial]] whose Galois group over <math>\mathbb{Q}</math> is the [[cyclic group]] {{math|'''Z'''/''n'''''Z'''}} for any positive [[integer]] {{mvar|n}}. To do this, choose a [[prime number|prime]] {{mvar|p}} such that {{math|''p'' ≡ 1 (mod ''n'')}}; this is possible by [[Dirichlet's theorem on arithmetic progressions|Dirichlet's theorem]]. Let {{math|'''Q'''(''μ'')}} be the [[Cyclotomic field#Cyclotomic fields|cyclotomic extension]] of <math>\mathbb{Q}</math> generated by {{mvar|μ}}, where {{mvar|μ}} is a primitive {{math|''p''}}-th [[root of unity]]; the Galois group of {{math|'''Q'''(''μ'')/'''Q'''}} is cyclic of order {{math|''p'' − 1}}.

Since {{mvar|n}} [[divisor|divides]] {{math|''p'' − 1}}, the Galois group has a cyclic [[subgroup]] {{mvar|H}} of order {{math|(''p'' − 1)/''n''}}. The [[fundamental theorem of Galois theory]] implies that the corresponding fixed field, {{math|1=''F'' = '''Q'''(''μ'')<sup>''H''</sup>}}, has Galois group {{math|'''Z'''/''n'''''Z'''}} over <math>\mathbb{Q}</math>. By taking appropriate sums of conjugates of {{mvar|μ}}, following the construction of [[Gaussian period]]s, one can find an element {{mvar|α}} of {{mvar|F}} that generates {{mvar|F}} over {{nowrap|<math>\mathbb{Q}</math>,}} and compute its [[Minimal polynomial (field theory)|minimal polynomial]].

This method can be extended to cover all finite [[abelian group]]s, since every such group appears in fact as a quotient of the Galois group of some cyclotomic extension of <math>\mathbb{Q}</math>. (This statement should not though be confused with the [[Kronecker–Weber theorem]], which lies significantly deeper.)

===Worked example: the cyclic group of order three===
For {{math|1=''n'' = 3}}, we may take {{math|1=''p'' = 7}}. Then {{math|Gal('''Q'''(''μ'')/'''Q''')}} is cyclic of order six. Let us take the generator {{mvar|η}} of this group which sends {{mvar|μ}} to {{math|''μ''<sup>3</sup>}}. We are interested in the subgroup {{math|1=''H'' = {1, ''η''<sup>3</sup>}}} of order two. Consider the element {{math|1=''α'' = ''μ'' + ''η''<sup>3</sup>(''μ'')}}. By construction, {{mvar|α}} is fixed by {{mvar|H}}, and only has three conjugates over <math>\mathbb{Q}</math>:

: {{math|1=''α'' = ''η''<sup>0</sup>(''α'') = ''μ'' + ''μ''<sup>6</sup>}}, 
: {{math|1=''β'' = ''η''<sup>1</sup>(''α'') = ''μ''<sup>3</sup> + ''μ''<sup>4</sup>}}, 
: {{math|1=''γ'' = ''η''<sup>2</sup>(''α'') = ''μ''<sup>2</sup> + ''μ''<sup>5</sup>}}.

Using the identity:

:{{math|1=1 + ''μ'' + ''μ''<sup>2</sup> + ⋯ + ''μ''<sup>6</sup> = 0}},

one finds that

: {{math|1=''α'' + ''β'' + ''γ'' = −1}},
: {{math|1=''αβ'' + ''βγ'' + ''γα'' = −2}},
: {{math|1=''αβγ'' = 1}}.

Therefore {{mvar|α}} is a [[root of a polynomial|root]] of the polynomial

:{{math|1=(''x'' − ''α'')(''x'' − ''β'')(''x'' − ''γ'') = ''x''<sup>3</sup> + ''x''<sup>2</sup> − 2''x'' − 1}},

which consequently has Galois group {{math|'''Z'''/3'''Z'''}} over <math>\mathbb{Q}</math>.

==Symmetric and alternating groups==
[[David Hilbert|Hilbert]] showed that all symmetric and [[alternating group|alternating]] groups are represented as Galois groups of polynomials with rational [[coefficient]]s.

The polynomial {{math|''x<sup>n</sup>'' + ''ax'' + ''b''}} has discriminant

:<math>(-1)^{\frac{n(n-1)}{2}} \!\left( n^n b^{n-1} + (-1)^{1-n} (n-1)^{n-1} a^n \right)\!.</math>

We take the special case

:{{math|1=''f''(''x'', ''s'') = ''x<sup>n</sup>'' − ''sx'' − ''s''}}.

Substituting a prime integer for {{mvar|s}} in {{math|''f''(''x'', ''s'')}} gives a polynomial (called a '''specialization''' of {{math|''f''(''x'', ''s'')}}) that by [[Eisenstein's criterion]] is [[irreducible polynomial|irreducible]]. Then {{math|''f''(''x'', ''s'')}} must be irreducible over <math>\mathbb{Q}(s)</math>. Furthermore, {{math|''f''(''x'', ''s'')}} can be written

:<math>x^n - \tfrac{x}{2} - \tfrac{1}{2} - \left( s - \tfrac{1}{2} \right)\!(x+1)</math>

and {{math|''f''(''x'', 1/2)}} can be factored to:

:<math>\tfrac{1}{2} (x-1)\!\left( 1+ 2x + 2x^2 + \cdots + 2 x^{n-1} \right)</math>

whose second factor is irreducible (but not by Eisenstein's criterion). Only the reciprocal polynomial is irreducible by Eisenstein's criterion. We have now shown that the group {{math|Gal(''f''(''x'', ''s'')/'''Q'''(''s''))}} is [[doubly transitive]].

We can then find that this Galois group has a transposition. Use the scaling {{math|1=(1 − ''n'')''x'' = ''ny''}} to get

:<math> y^n - \left \{ s \left ( \frac{1-n}{n} \right )^{n-1} \right \} y - \left \{ s \left ( \frac{1-n}{n} \right )^n \right \}</math>

and with

:<math> t = \frac{s (1-n)^{n-1}}{n^n},</math>

we arrive at:

:{{math|1=''g''(''y'', ''t'') = ''y<sup>n</sup>'' − ''nty'' + (''n'' − 1)''t''}}

which can be arranged to

:{{math|''y<sup>n</sup>'' − ''y'' − (''n'' − 1)(''y'' − 1) + (''t'' − 1)(−''ny'' + ''n'' − 1)}}.

Then {{math|''g''(''y'', 1)}} has {{math|1}} as a [[double root|double zero]] and its other {{math|''n'' − 2}} zeros are [[simple zero|simple]], and a transposition in {{math|Gal(''f''(''x'', ''s'')/'''Q'''(''s''))}} is implied. Any finite [[doubly transitive permutation group]] containing a transposition is a full symmetric group.

[[Hilbert's irreducibility theorem]] then implies that an infinite set of rational numbers give specializations of {{math|''f''(''x'', ''t'')}} whose Galois groups are {{math|''S<sub>n</sub>''}} over the rational field {{nowrap|<math>\mathbb{Q}</math>.}} In fact this set of rational numbers is dense in {{nowrap|<math>\mathbb{Q}</math>.}}

The discriminant of {{math|''g''(''y'', ''t'')}} equals

:<math> (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} (1-t),</math>

and this is not in general a perfect square.

===Alternating groups===
Solutions for alternating groups must be handled differently for [[parity (mathematics)|odd]] and [[parity (mathematics)|even]] degrees.

====Odd degree====
Let

:<math>t = 1 - (-1)^{\tfrac{n(n-1)}{2}} n u^2</math>

Under this substitution the discriminant of {{math|''g''(''y'', ''t'')}} equals

:<math>\begin{align}
(-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} (1-t)
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left (1 - \left (1 - (-1)^{\tfrac{n(n-1)}{2}} n u^2 \right ) \right) \\
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left ((-1)^{\tfrac{n(n-1)}{2}} n u^2 \right ) \\
&= n^{n+1} (n-1)^{n-1} t^{n-1} u^2 
\end{align}</math>

which is a perfect square when {{mvar|n}} is odd.

====Even degree====
Let:

:<math>t = \frac{1}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2}</math>

Under this substitution the discriminant of {{math|''g''(''y'', ''t'')}} equals:

:<math>\begin{align}
(-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} (1-t)
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left (1 - \frac{1}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2} \right ) \\
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left ( \frac{\left ( 1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2 \right ) - 1}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2} \right ) \\
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left ( \frac{(-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2} \right ) \\
&= (-1)^{\frac{n(n-1)}{2}} n^n (n-1)^{n-1} t^{n-1} \left (t (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2 \right ) \\
&= n^n (n-1)^n t^n u^2
\end{align}</math>

which is a perfect square when {{mvar|n}} is even.

Again, Hilbert's irreducibility theorem implies the existence of infinitely many specializations whose Galois groups are alternating groups.

==Rigid groups==
Suppose that {{math|''C''<sub>1</sub>, …, ''C<sub>n</sub>''}} are [[conjugacy class]]es of a finite group {{mvar|G}}, and {{mvar|A}} be the set of {{mvar|n}}-tuples {{math|(''g''<sub>1</sub>, …, ''g<sub>n</sub>'')}} of {{mvar|G}} such that {{math|''g<sub>i</sub>''}} is in {{math|''C<sub>i</sub>''}} and the product {{math|''g''<sub>1</sub>…''g<sub>n</sub>''}} is trivial. Then {{mvar|A}} is called '''rigid''' if it is [[empty set|nonempty]], {{mvar|G}} acts transitively on it by conjugation, and each element of {{mvar|A}} generates {{mvar|G}}.

{{harvtxt|Thompson|1984}} showed that if a finite group {{mvar|G}} has a rigid set then it can often be realized as a Galois group over a cyclotomic extension of the rationals. (More precisely, over the cyclotomic extension of the rationals generated by the values of the irreducible characters of {{mvar|G}} on the conjugacy classes {{math|''C<sub>i</sub>''}}.)

This can be used to show that many finite simple groups, including the [[monster group]], are Galois groups of extensions of the rationals. The monster group is generated by a triad of elements of orders {{math|2}}, {{math|3}}, and {{math|29}}. All such triads are conjugate.

The prototype for rigidity is the symmetric group {{math|''S<sub>n</sub>''}}, which is generated by an {{mvar|n}}-cycle and a transposition whose product is an {{math|(''n'' − 1)}}-cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group.

==A construction with an elliptic modular function==
Let {{math|''n'' > 1}} be any integer. A lattice {{math|Λ}} in the [[complex plane]] with period ratio {{mvar|τ}} has a sublattice {{math|Λ′}} with period ratio {{math|''nτ''}}. The latter lattice is one of a finite set of sublattices permuted by the [[modular group]] {{math|PSL(2, '''Z''')}}, which is based on changes of basis for {{math|Λ}}. Let {{mvar|j}} denote the [[elliptic modular function]] of [[Felix Klein]]. Define the polynomial {{math|''φ<sub>n</sub>''}} as the product of the differences {{math|(''X'' − ''j''(Λ<sub>''i''</sub>))}} over the conjugate sublattices. As a polynomial in {{mvar|X}}, {{math|''φ<sub>n</sub>''}} has coefficients that are polynomials over <math>\mathbb{Q}</math> in {{math|''j''(''τ'')}}.

On the conjugate lattices, the modular group acts as {{math|PGL(2, '''Z'''/''n'''''Z''')}}. It follows that {{math|''φ<sub>n</sub>''}} has Galois group isomorphic to {{math|PGL(2, '''Z'''/''n'''''Z''')}} over <math>\mathbb{Q}(\mathrm{J}(\tau))</math>.

Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing {{math|''φ<sub>n</sub>''}} to polynomials with Galois group {{math|PGL(2, '''Z'''/''n'''''Z''')}} over {{nowrap|<math>\mathbb{Q}</math>.}} The groups {{math|PGL(2, '''Z'''/''n'''''Z''')}} include infinitely many non-solvable groups.
== See also ==
* [[Semiabelian group (Galois theory)]]

==Notes==
{{Reflist}}

== References ==
*{{cite journal |doi=10.1112/BLMS/1.3.332|title=Extensions of the Rationals with Galois Group PGL(2,Z<sub>n</sub>) |year=1969 |last1=MacBeath |first1=A. M. |journal=Bulletin of the London Mathematical Society |volume=1 |issue=3 |pages=332–338 }}
*{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Some finite groups which appear as Gal L/K, where K ⊆ Q(μ<sub> n</sub>) |mr=751155 | year=1984 | journal=Journal of Algebra | volume=89 | issue=2 | pages=437–499 | doi=10.1016/0021-8693(84)90228-X| doi-access=free }}
* Helmut Völklein, ''Groups as Galois Groups, an Introduction'', Cambridge University Press, 1996. ISBN 978-0521065030 . 
* {{cite book | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | title=Topics in Galois Theory | series=Research Notes in Mathematics | volume=1 | publisher=Jones and Bartlett | year=1992 | isbn=0-86720-210-6 | zbl=0746.12001 }}
* Gunter Malle, Heinrich Matzat, ''Inverse Galois Theory'', Springer-Verlag, 1999, {{ISBN|3-540-62890-8}}.
* Gunter Malle, Heinrich Matzat, ''Inverse Galois Theory'', 2nd edition, Springer-Verlag, 2018.
* Alexander Schmidt, Kay Wingberg, ''[https://web.archive.org/web/20050830000818/http://www.math.uiuc.edu/Algebraic-Number-Theory/0136/ Safarevic's Theorem on Solvable Groups as Galois Groups]'' (''see also'' {{Neukirch et al. CNF}})
* Christian U. Jensen, Arne Ledet, and [[Noriko Yui]], ''Generic Polynomials, Constructive Aspects of the Inverse Galois Problem'', Cambridge University Press, 2002.

==External links==
*{{cite web |title=Inverse Galois Problem PCMI 2021 Graduate Summer School Program - Number Theory Informed by Computation - July 26-30, 2021 |url=https://people.maths.bris.ac.uk/~matyd/InvGal/|archive-url=https://web.archive.org/web/20230216095734/https://people.maths.bris.ac.uk/~matyd/InvGal/|archive-date=2023-02-16 }}
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[[Category:Galois theory]]
[[Category:Unsolved problems in mathematics]]