Inverse Galois problem

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In mathematics, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers Q. This problem, first posed in the 19th century[citation needed], is unsolved.

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 isomorphic to G? One says that G is realizable over K if such a field L exists.

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[edit] Partial results

There is a great deal of detailed information in particular cases. It is known that every finite group is realizable over any 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 zero. Shafarevich showed that every finite solvable group is realizable over Q.

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 field KG 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 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 G geometrically as a Galois covering of the projective line: in algebraic terms, start with an extension of the field Q(t) of rational functions 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.

[edit] A simple example: cyclic groups

It is possible, using classical results, to construct explicitly a polynomial whose Galois group over Q is the cyclic group

Z/nZ

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. Let

Q(μ)

be the cyclotomic extension of Q generated by μ, where μ is a primitive pth root of unity; the Galois group of

Q(μ)/Q

is cyclic of order

p − 1.

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

F = Q(μ)H

has Galois group

Z/nZ

over Q. By taking appropriate sums of conjugates of μ, following the construction of Gaussian periods, one can find an element α of F that generates F over Q, and to 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 Q. (This statement should not though be confused with the Kronecker-Weber theorem, which lies significantly deeper.)

[edit] Worked example: the cyclic group of order three

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 μ3. We are interested in the subgroup H = {1, η3} of order two. Consider the element α = μ + η3(μ). By construction, α is fixed by H, and only has three conjugates over Q, given by

α = μ + μ6,    β = η(α) = μ3 + μ4,    γ = η2(α) = μ2 + μ5.

Using the identity 1 + μ + μ2 + ... + μ6 = 0, one finds that

α + β + γ = −1,
αβ + βγ + γα = −2, and
αβγ = 1.

Therefore α is a root of the polynomial

(x − α)(x − β)(x − γ) = x3 + x2 − 2x − 1,

which consequently has Galois group Z/3Z over Q.

[edit] Symmetric and alternating groups

Hilbert showed that all symmetric and alternating groups are represented as Galois groups of polynomials with rational coefficients.

The polynomial xn+ax+b has discriminant

(-1)n(n-1)/2[nnbn-1 + (-1)1-n(n-1)n-1an].

We take the special case

f(x,s) = xn - sx - s

Substituting a prime integer for t 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

xn - x/2 - 1/2 - (s-1/2)(x+1)

and f(x,1/2) can be factored to:

(x-1)(1 + 2x + 2x2 +... + 2xn-1)/2

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.

We can then find that this Galois group has a tranposition. Use the scaling (1-n)x=ny to get

yn - s((1-n)/n)n-1y - s((1-n)/n)n

and with t=s(1-n)n-1/nn get

g(y,t) = yn - nty + (n-1)t

which can be arranged to

yn - y + (n-1)(y-1) + (t-1)(-ny+n-1).

Then g(y,1) has 1 as a 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.

Hilbert's irreducibility theorem then implies that an infinite set of rational numbers give specializations of f(x,t) whose Galois groups are Sn over the rational field Q. In fact this set of rational numbers is dense in Q.

The discriminant of g(y,t) equals

(-1)n(n-1)/2nn(n-1)n-1tn-1(1-t)

and this is not in general a perfect square.

[edit] Alternating groups

Solutions for alternating groups must be handled differently for odd and even degrees. Let

t = 1 - (-1)n(n-1)/2nu2

Under this substitution the discriminant of g(y,t) equals

nn+1(n-1)n-1tn-1u2

which is a perfect square when n is odd.

In the even case let t be the reciprocal of

1 + (-1)n(n-1)/2(n-1)u2

and 1-t becomes

t(-1)n(n-1)/2(n-1)u2

and the discriminant becomes

nn(n-1)ntnu2

which is a perfect square when n is even.

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

[edit] Rigid groups

Suppose that C1,...,Cn are conjugacy classes of a finite group G, and A be the set of n-tuples (g1,...gn) of G such that gi is in Ci and the product g1...gn 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.

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 Ci.)

This can be used to show that many finite simple groups, including the monster simple group, are Galois groups of extensions of the rationals.

The prototype for rigidity is the symmetric group Sn, 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.

[edit] References

  • Thompson, John G. (1984), “Some finite groups which appear as Gal L/K, where K⊆ Q(μ n)”, Journal of Algebra 89 (2): 437–499, MR751155, ISSN 0021-8693 
  • Helmut Völklein, Groups as Galois Groups, an Introduction, Cambridge University Press, 1996.
  • Gunter Malle, Heinrich Matzat, Inverse Galois Theory, Springer-Verlag, 1999, ISBN 3-540-62890-8.
  • Alexander Schmidt, Kay Wingberg, Safarevic's Theorem on Solvable Groups as Galois Groups (see also Jürgen Neukirch, Alexander Schmidt, Kay Wingberg, Cohomology of Number Fields, Springer-Verlag, 1999, ISBN 3-540-66671-0.)
  • 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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