There are some permutation groups for which generic polynomials are known, which define all algebraic extensionsof 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 order8.
More generally, let G be a given finite group, and K a field. If there is a Galois extension field L/K whose Galois group is isomorphictoG, one says that G is realizable over K.
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 ofrational 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.
All permutation groups of degree 16 or less are known to be realizable over ;[4] the group PSL(2,16):2 of degree 17 may not be.[5]
All 13 non-abeliansimple groups smaller than PSL(2,25) (order 7800) are known to be realizable over .[6]
It is possible, using classical results, to construct explicitly a polynomial whose Galois group over is the cyclic groupZ/nZ for any positive integern. To do this, choose a primep such that p ≡ 1 (mod n); this is possible by Dirichlet's theorem. Let Q(μ) be the cyclotomic extensionof generated by μ, where μ is a primitive p-th root of unity; the Galois group of Q(μ)/Q is cyclic of order p − 1.
Since ndividesp − 1, the Galois group has a cyclic subgroupH 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 . By taking appropriate sums of conjugates of μ, following the construction of Gaussian periods, one can find an element αofF that generates F over , 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 . (This statement should not though be confused with the Kronecker–Weber theorem, which lies significantly deeper.)
Worked example: the cyclic group of order three[edit]
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 μtoμ3. 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 :
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
We take the special case
f(x, s) = xn − sx − s.
Substituting a prime integer for sinf(x, s) gives a polynomial (called a specializationoff(x, s)) that by Eisenstein's criterionisirreducible. Then f(x, s) must be irreducible over . Furthermore, f(x, s) can be written
and f(x, 1/2) can be factored to:
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 Gal(f(x, s)/Q(s))isdoubly transitive.
We can then find that this Galois group has a transposition. Use the scaling (1 − n)x = ny to get
and with
we arrive at:
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 . In fact this set of rational numbers is dense in .
Suppose that C1, …, Cn are conjugacy classes of a finite group G, and A be the set of n-tuples (g1, …, gn)ofG 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 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.
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.
A construction with an elliptic modular function[edit]
Let n >1 be any integer. 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 groupPSL(2, Z), which is based on changes of basis for Λ. Let j denote the elliptic modular functionofFelix Klein. Define the polynomial φn as the product of the differences (X − j(Λi)) over the conjugate sublattices. As a polynomial in X, φn has coefficients that are polynomials over inj(τ).
On the conjugate lattices, the modular group acts as PGL(2, Z/nZ). It follows that φn has Galois group isomorphic to PGL(2, Z/nZ) over .
Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing φn to polynomials with Galois group PGL(2, Z/nZ) over . The groups PGL(2, Z/nZ) include infinitely many non-solvable groups.
MacBeath, A. M. (1969). "Extensions of the Rationals with Galois Group PGL(2,Zn)". Bulletin of the London Mathematical Society. 1 (3): 332–338. doi:10.1112/BLMS/1.3.332.
Christian U. Jensen, Arne Ledet, and Noriko Yui, Generic Polynomials, Constructive Aspects of the Inverse Galois Problem, Cambridge University Press, 2002.