There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of
It is known that every finite group is realizable over any function field in one variable over the complex numbers
Igor Shafarevich showed that every finite solvable group is realizable over
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
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.
[5] All 13 non-abelian simple 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 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.
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
: Using the identity: one finds that Therefore α is a root of the polynomial which consequently has Galois group Z/3Z over
The polynomial xn + ax + b has discriminant We take the special case 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.
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
Solutions for alternating groups must be handled differently for odd and even degrees.
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.)
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.
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.
Define the polynomial φn as the product of the differences (X − j(Λi)) over the conjugate sublattices.
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