Hilbertian Fields

David Hilbert proved his celebrated irreducibility theorem during his attempt to solve a central problem of Galois theory: Is every finite group realizable over ℚ? He proved that a general specialization of the coeficients of the general polynomial of degree n to elements of ℚ gives a polynomial whose Galois group is S _n . Further, if f∈ℚ[T₁,…,T _r ,X] is an irreducible polynomial, then there exist a₁,…,a _r ∈Q such that f(a,X) remains irreducible. This result is now known as Hilbert’s irreducibility theorem. Since then, many more finite groups have been realized over ℚ. Most of those have been realized via Hilbert’s theorem. This has brought the theorem to the center of the theory of fields.