Hilbert's Irreducibility Theorem - Formulation of The Theorem

Formulation of The Theorem

Hilbert's irreducibility theorem. Let

be irreducible polynomials in the ring

Then there exists an r-tuple of rational numbers (a₁,...,a_r) such that

are irreducible in the ring

Remarks.

• It follows from the theorem that there are infinitely many r-tuples. In fact the set of all irreducible specialization, called Hilbert set, is large in many senses. For example, this set is Zariski dense in

• There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (a₁,...,a_r) to be integers.

• There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, global fields are Hilbertian.

• The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take in the definition. A recent result of Bary-Soroker shows that for a field K to be Hilbertian it suffices to consider the case of and absolutely irreducible, that is, irreducible in the ring Kalg, where Kalg is the algebraic closure of K.