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| 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 '''Q'''. This problem, first posed in the 19th century,<ref>http://udini.proquest.com/view/the-inverse-galois-problem-and-pqid:2439411211</ref><!--The reference text reads "In 1892 Hilbert [13, 21] proposed the first systematic approach to solving the question"--> is unsolved.
| | To reach it in excel, copy-paste this continued plan easily into corpuscle B1. A person have again access an majority of time in abnormal located in corpuscle A1, the standard in treasures will come to the forefront in B1.<br><br> |
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| There are some permutation groups for which [[generic polynomial]]s are known, which define all algebraic extensions of Q having a particular group as Galois group. These groups include all of degree no greater than 5. There also are groups known not to have generic polynomials, such as the cyclic group of order 8.
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| More generally, let ''G'' be a given finite group, and let ''K'' be a field. Then the question is this: is there a [[Galois extension]] field ''L''/''K'' such that the Galois group of the extension is [[group isomorphism|isomorphic]] to ''G''? One says that '''G is realizable over K''' if such a field ''L'' exists.
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| ==Partial results==
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| There is a great deal of detailed information in particular cases. 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 numbers]] '''C''', and more generally over function fields in one variable over any [[algebraically closed]] field of [[characteristic (algebra)|characteristic]] zero. [[Shafarevich]] showed that every finite [[solvable group]] is realizable over '''Q'''.<ref>I.R. Shafarevich, ''The imbedding problem for splitting extensions'', Dokl. Akad.
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| Nauk SSSR '''120''' (1958), 1217-1219.</ref> It is also known that every [[sporadic group]], except possibly the [[Mathieu group]] M<sub>23</sub>, is realizable over '''Q'''.<ref>p. 5 of Jensen et al., 2002</ref>
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| [[David Hilbert|Hilbert]] had shown that this question is related to a [[rationality question]] for ''G'': if ''K'' is any extension of '''Q''', on which ''G'' acts as an [[automorphism group]] and the [[Invariant theory|invariant field]] ''K<sup>G</sup>'' is rational over '''Q''', then ''G'' is realizable over '''Q'''. Here ''rational'' means that it is a [[purely transcendental]] extension of '''Q''', generated by an [[algebraically independent]] set. This criterion can for example be used to show that all the [[symmetric group]]s are realizable.
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| 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 ''G'' geometrically as a [[Galois covering]] of the [[projective line]]: in algebraic terms, starting with an extension of the field '''Q'''(''t'') of [[rational function]]s in an indeterminate ''t''. After that, one applies [[Hilbert's irreducibility theorem]] to specialise ''t'', in such a way as to preserve the Galois group.
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| ==A simple example: cyclic groups==
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| It is possible, using classical results, to construct explicitly a polynomial whose Galois group over '''Q''' is the [[cyclic group]] '''Z'''/''n'''''Z''' for any positive integer ''n''. To do this, choose a prime ''p'' such that ''p'' ≡ 1 (mod ''n''); this is possible by [[Dirichlet's theorem on arithmetic progressions|Dirichlet's theorem]]. Let '''Q'''(μ) be the [[Cyclotomic field#Cyclotomic fields|cyclotomic extension]] of '''Q''' generated by μ, where μ is a primitive ''p''<sup>th</sup> [[root of unity]]; the Galois group of '''Q'''(μ)/'''Q''' is cyclic of order ''p'' − 1.
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| Since ''n'' divides ''p'' − 1, the Galois group has a cyclic subgroup ''H'' of order (''p'' − 1)/''n''. The [[fundamental theorem of Galois theory]] implies that the corresponding fixed field
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| :<math> F = \bold Q(\mu)^H </math>
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| has Galois group '''Z'''/''n'''''Z''' over '''Q'''. By taking appropriate sums of conjugates of μ, following the construction of [[Gaussian period]]s, one can find an element α of ''F'' that generates ''F'' over '''Q''', and compute its minimal polynomial.
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| 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 '''Q'''. (This statement should not though be confused with the [[Kronecker–Weber theorem]], which lies significantly deeper.)
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| ===Worked example: the cyclic group of order three===
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| For ''n'' = 3, we may take ''p'' = 7. Then Gal('''Q'''(μ)/'''Q''') is cyclic of order six. Let us take the generator η of this group which sends μ to μ<sup>3</sup>. We are interested in the subgroup ''H'' = {1, η<sup>3</sup>} of order two. Consider the element α = μ + η<sup>3</sup>(μ). By construction, α is fixed by ''H'', and only has three conjugates over '''Q''', given by
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| : α = μ + μ<sup>6</sup>, β = η(α) = μ<sup>3</sup> + μ<sup>4</sup>, γ = η<sup>2</sup>(α) = μ<sup>2</sup> + μ<sup>5</sup>. | |
| Using the identity 1 + μ + μ<sup>2</sup> + ... + μ<sup>6</sup> = 0, one finds that
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| : α + β + γ = −1,
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| : αβ + βγ + γα = −2, and
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| : αβγ = 1.
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| Therefore α is a root of the polynomial
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| : (''x'' − α)(''x'' − β)(''x'' − γ) = ''x''<sup>3</sup> + ''x''<sup>2</sup> − 2''x'' − 1,
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| which consequently has Galois group '''Z'''/3'''Z''' over '''Q'''.
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| ==Symmetric and alternating groups==
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| [[David Hilbert|Hilbert]] showed that all symmetric and alternating groups are represented as Galois groups of polynomials with rational coefficients.
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| The polynomial {{nowrap|''x<sup>n</sup>'' + ''ax'' + ''b''}} has discriminant
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| :(−1)<sup>''n''(''n''−1)/2</sup>[''n<sup>n</sup>b''<sup>''n''−1</sup> + (−1)<sup>1−''n''</sup>(''n'' − 1)<sup>''n''−1</sup>''a<sup>n</sup>''].
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| We take the special case
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| :''f''(''x'',''s'') = ''x<sup>n</sup>'' − ''sx'' − ''s''.
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| Substituting a prime integer for ''s'' in ''f''(''x'',''s'') gives a polynomial (called a '''specialization''' of ''f''(''x'',''s'')) that by [[Eisenstein's criterion]] is irreducible. Then ''f''(''x'',''s'') must be irreducible over '''Q'''(''s''). Furthermore, ''f''(''x'',''s'') can be written
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| :''x<sup>n</sup>'' − ''x''/2 − 1/2 − (''s'' − 1/2)(''x'' + 1)
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| and ''f''(''x'',1/2) can be factored to: | |
| :(''x'' − 1)(1 + 2''x'' + 2''x''<sup>2</sup> + ... + 2''x''<sup>''n''−1</sup>)/2
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| whose second factor is irreducible by Eisenstein's criterion. We have now shown that the group Gal(''f''(''x'',''s'')/'''Q'''(''s'')) is [[doubly transitive]].
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| We can then find that this Galois group has a transposition. Use the scaling {{nowrap|(1 − ''n'')''x'' {{=}} ''ny''}} to get
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| :''y<sup>n</sup>'' − ''s''((1 − ''n'')/''n'')<sup>''n''−1</sup>''y'' − ''s''((1 − ''n'')/''n'')<sup>''n''</sup>
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| and with {{nowrap|''t'' {{=}} ''s''(1 − ''n'')<sup>''n''−1</sup>/''n<sup>n</sup>''}} get
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| :''g''(''y'',''t'') = ''y<sup>n</sup>'' − ''nty'' + (''n'' − 1)''t''
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| which can be arranged to
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| :''y<sup>n</sup>'' − ''y'' − (''n'' − 1)(''y'' − 1) + (''t'' − 1)(−''ny'' + ''n'' − 1).
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| Then ''g''(''y'',1) has 1 as a [[simple zero|double zero]] and its other ''n'' − 2 zeros are simple, and a transposition in Gal(''f''(''x'',''s'')/'''Q'''(''s'')) is implied. Any finite [[doubly transitive permutation group]] containing a transposition is a full symmetric group.
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| [[Hilbert's irreducibility theorem]] then implies that an infinite set of rational numbers give specializations of ''f''(''x'',''t'') whose Galois groups are S<sub>''n''</sub> over the rational field '''Q'''. In fact this set of rational numbers is dense in '''Q'''.
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| The discriminant of ''g''(''y'',''t'') equals
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| :(−1)<sup>''n''(''n''−1)/2</sup>''n<sup>n</sup>''(''n'' − 1)<sup>''n''−1</sup>''t''<sup>''n''−1</sup>(1 − ''t'')
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| and this is not in general a perfect square.
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| ===Alternating groups===
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| Solutions for alternating groups must be handled differently for odd and even degrees.
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| In the odd case, let
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| :''t'' = 1 − (−1)<sup>''n''(''n''−1)/2</sup>''nu''<sup>2</sup>
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| Under this substitution the discriminant of ''g''(''y'',''t'') equals
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| :''n''<sup>''n''+1</sup>(''n'' − 1)<sup>''n''−1</sup>''t''<sup>''n''−1</sup>''u''<sup>2</sup>
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| which is a perfect square when ''n'' is odd.
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| In the even case let t be the reciprocal of
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| :1 + (−1)<sup>''n''(''n''−1)/2</sup>(''n'' − 1)''u''<sup>2</sup>
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| and 1 − ''t'' becomes
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| :''t''(−1)<sup>''n''(''n''−1)/2</sup>(''n'' − 1)''u''<sup>2</sup>
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| and the discriminant becomes
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| :''n<sup>n</sup>''(''n'' − 1)<sup>''n''</sup>''t<sup>n</sup>u''<sup>2</sup>
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| which is a perfect square when ''n'' is even.
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| Again, Hilbert's irreducibility theorem implies the existence of infinitely many specializations whose Galois groups are alternating groups.
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| ==Rigid groups==
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| Suppose that ''C''<sub>1</sub>,...,''C''<sub>''n''</sub> are conjugacy classes of a finite group ''G'',
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| and ''A'' be the set of ''n''-tuples (''g''<sub>1</sub>,...''g''<sub>''n''</sub>) of ''G'' such that ''g''<sub>''i''</sub> is in ''C''<sub>''i''</sub> and the product ''g''<sub>1</sub>...''g''<sub>''n''</sub> is trivial. Then ''A'' is called '''rigid''' if it is nonempty, ''G'' acts transitively on it by conjugation, and each element of ''A'' generates ''G''.
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| {{harvtxt|Thompson|1984}} showed that if a finite group ''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 ''G'' on the conjugacy classes ''C''<sub>''i''</sub>.)
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| 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 2, 3, and 29. All such triads are conjugate.
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| The prototype for rigidity is the symmetric group S<sub>n</sub>, which is generated by an n-cycle and a transposition whose product is an (n-1)-cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group. | |
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| ==A construction with an elliptic modular function==
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| Let n be any integer greater than 1. A lattice Λ in the complex plane with period ratio τ has a sublattice Λ' with period ratio nτ. The latter lattice is one of a finite set of sublattices permuted by the [[modular group]] PSL(2,'''Z'''), which is based on changes of basis for Λ. Let j denote the [[elliptic modular function]] of Klein. Define the polynomial φ<sub>n</sub> as the product of the differences (''X''-j(Λ<sub>i</sub>)) over the conjugate sublattices. As a polynomial in ''X'', φ<sub>n</sub> has coefficients that are polynomials over '''Q''' in j(τ).
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| On the conjugate lattices, the modular group acts as PGL(2,'''Z'''<sub>n</sub>). It follows that φ<sub>n</sub> has Galois group isomorphic to PGL(2,'''Z'''<sub>n</sub>) over '''Q'''(J(τ)).
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| Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing φ<sub>n</sub> to polynomials with Galois group PGL(2,'''Z'''<sub>n</sub>) over '''Q'''. The groups PGL(2,'''Z'''<sub>n</sub>) include infinitely many non-solvable groups.
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| ==Notes==
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| <references/>
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| == References ==
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| * Alexander M. Macbeath, ''Extensions of the Rationals with Galois Group PGL(2,Z<sub>n</sub>)'', Bull. London Math. Soc., 1 (1969),332-338.
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| *{{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>) | id={{MathSciNet | id = 751155}} | year=1984 | journal=Journal of Algebra | volume=89 | issue=2 | pages=437–499 | doi=10.1016/0021-8693(84)90228-X}}
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| * Helmut Völklein, ''Groups as Galois Groups, an Introduction'', Cambridge University Press, 1996.
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| * {{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 }}
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| * Gunter Malle, Heinrich Matzat, ''Inverse Galois Theory'', Springer-Verlag, 1999, ISBN 3-540-62890-8.
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| * Alexander Schmidt, Kay Wingberg, ''[http://www.math.uiuc.edu/Algebraic-Number-Theory/0136/ Safarevic's Theorem on Solvable Groups as Galois Groups]'' (''see also'' {{Neukirch et al. CNF}})
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| * Christian U. Jensen, Arne Ledet, and [[Noriko Yui]], ''Generic Polynomials, Constructive Aspects of the Inverse Galois Problem'', Cambridge University Press, 2002.
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| [[Category:Galois theory]]
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| [[Category:Unsolved problems in mathematics]]
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