Template:Short description Template:Unsolved 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 numbers <math>\mathbb{Q}</math>. This problem, first posed in the early 19th century,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> is unsolved.
There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of <math>\mathbb{Q}</math> having a particular group as Galois group. These groups include all of degree no greater than Template:Math. There also are groups known not to have generic polynomials, such as the cyclic group of order Template:Math.
More generally, let Template:Mvar be a given finite group, and Template:Mvar a field. If there is a Galois extension field Template:Math whose Galois group is isomorphic to Template:Mvar, one says that Template:Mvar is realizable over Template:Mvar.
Partial resultsEdit
Many cases are known. It is known that every finite group is realizable over any function field in one variable over the complex numbers <math>\mathbb{C}</math>, and more generally over function fields in one variable over any algebraically closed field of 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 Template:Math, 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 Template:Mvar:
- If Template:Mvar is any extension of <math>\mathbb{Q}</math> on which Template:Mvar acts as an automorphism group, and the invariant field Template:Math is rational over Template:Nowrap then Template:Mvar is realizable over Template:Nowrap
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 groups 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 Template:Mvar 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 functions in an indeterminate Template:Mvar. After that, one applies Hilbert's irreducibility theorem to specialise Template:Mvar, in such a way as to preserve the Galois group.
All permutation groups of degree 23 or less, except the Mathieu group Template:Math, are known to be realizable over Template:Nowrap.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
All 13 non-abelian simple groups smaller than PSL(2,25) (order 7800) are known to be realizable over Template:Nowrap<ref>Malle and Matzat (1999), pp. 403-424</ref>
A simple example: cyclic groupsEdit
It is possible, using classical results, to construct explicitly a polynomial whose Galois group over <math>\mathbb{Q}</math> is the cyclic group Template:Math for any positive integer Template:Mvar. To do this, choose a prime Template:Mvar such that Template:Math; this is possible by Dirichlet's theorem. Let Template:Math be the cyclotomic extension of <math>\mathbb{Q}</math> generated by Template:Mvar, where Template:Mvar is a primitive Template:Math-th root of unity; the Galois group of Template:Math is cyclic of order Template:Math.
Since Template:Mvar divides Template:Math, the Galois group has a cyclic subgroup Template:Mvar of order Template:Math. The fundamental theorem of Galois theory implies that the corresponding fixed field, Template:Math, has Galois group Template:Math over <math>\mathbb{Q}</math>. By taking appropriate sums of conjugates of Template:Mvar, following the construction of Gaussian periods, one can find an element Template:Mvar of Template:Mvar that generates Template:Mvar over Template:Nowrap and compute its minimal polynomial.
This method can be extended to cover all finite abelian groups, 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 threeEdit
For Template:Math, we may take Template:Math. Then Template:Math is cyclic of order six. Let us take the generator Template:Mvar of this group which sends Template:Mvar to Template:Math. We are interested in the subgroup Template:Math} of order two. Consider the element Template:Math. By construction, Template:Mvar is fixed by Template:Mvar, and only has three conjugates over <math>\mathbb{Q}</math>:
Using the identity:
one finds that
Therefore Template:Mvar is a root of the polynomial
which consequently has Galois group Template:Math over <math>\mathbb{Q}</math>.
Symmetric and alternating groupsEdit
Hilbert showed that all symmetric and alternating groups are represented as Galois groups of polynomials with rational coefficients.
The polynomial Template:Math 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
Substituting a prime integer for Template:Mvar in Template:Math gives a polynomial (called a specialization of Template:Math) that by Eisenstein's criterion is irreducible. Then Template:Math must be irreducible over <math>\mathbb{Q}(s)</math>. Furthermore, Template:Math can be written
- <math>x^n - \tfrac{x}{2} - \tfrac{1}{2} - \left( s - \tfrac{1}{2} \right)\!(x+1)</math>
and Template:Math 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 Template:Math is doubly transitive.
We can then find that this Galois group has a transposition. Use the scaling Template:Math 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:
which can be arranged to
Then Template:Math has Template:Math as a double zero and its other Template:Math zeros are simple, and a transposition in Template:Math 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 Template:Math whose Galois groups are Template:Math over the rational field Template:Nowrap In fact this set of rational numbers is dense in Template:Nowrap
The discriminant of Template:Math 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 groupsEdit
Solutions for alternating groups must be handled differently for odd and even degrees.
Odd degreeEdit
Let
- <math>t = 1 - (-1)^{\tfrac{n(n-1)}{2}} n u^2</math>
Under this substitution the discriminant of Template:Math 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 Template:Mvar is odd.
Even degreeEdit
Let:
- <math>t = \frac{1}{1 + (-1)^{\tfrac{n(n-1)}{2}} (n-1) u^2}</math>
Under this substitution the discriminant of Template:Math 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 Template:Mvar is even.
Again, Hilbert's irreducibility theorem implies the existence of infinitely many specializations whose Galois groups are alternating groups.
Rigid groupsEdit
Suppose that Template:Math are conjugacy classes of a finite group Template:Mvar, and Template:Mvar be the set of Template:Mvar-tuples Template:Math of Template:Mvar such that Template:Math is in Template:Math and the product Template:Math is trivial. Then Template:Mvar is called rigid if it is nonempty, Template:Mvar acts transitively on it by conjugation, and each element of Template:Mvar generates Template:Mvar.
Template:Harvtxt showed that if a finite group Template:Mvar 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 Template:Mvar on the conjugacy classes Template:Math.)
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 Template:Math, Template:Math, and Template:Math. All such triads are conjugate.
The prototype for rigidity is the symmetric group Template:Math, which is generated by an Template:Mvar-cycle and a transposition whose product is an Template:Math-cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group.
A construction with an elliptic modular functionEdit
Let Template:Math be any integer. A lattice Template:Math in the complex plane with period ratio Template:Mvar has a sublattice Template:Math with period ratio Template:Math. The latter lattice is one of a finite set of sublattices permuted by the modular group Template:Math, which is based on changes of basis for Template:Math. Let Template:Mvar denote the elliptic modular function of Felix Klein. Define the polynomial Template:Math as the product of the differences Template:Math over the conjugate sublattices. As a polynomial in Template:Mvar, Template:Math has coefficients that are polynomials over <math>\mathbb{Q}</math> in Template:Math.
On the conjugate lattices, the modular group acts as Template:Math. It follows that Template:Math has Galois group isomorphic to Template:Math over <math>\mathbb{Q}(\mathrm{J}(\tau))</math>.
Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing Template:Math to polynomials with Galois group Template:Math over Template:Nowrap The groups Template:Math include infinitely many non-solvable groups.
See alsoEdit
NotesEdit
ReferencesEdit
- Template:Cite journal
- Template:Citation
- Helmut Völklein, Groups as Galois Groups, an Introduction, Cambridge University Press, 1996. ISBN 978-0521065030 .
- Template:Cite book
- Gunter Malle, Heinrich Matzat, Inverse Galois Theory, Springer-Verlag, 1999, Template:ISBN.
- Gunter Malle, Heinrich Matzat, Inverse Galois Theory, 2nd edition, Springer-Verlag, 2018.
- Alexander Schmidt, Kay Wingberg, Safarevic's Theorem on Solvable Groups as Galois Groups (see also Template: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 linksEdit
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