Hilbert's irreducibility theorem

Formulation of the theorem

• It follows from the theorem that there are infinitely many r-tuples. In fact the set of all irreducible specializations, called Hilbert set, is large in many senses. For example, this set is Zariski dense in
• There are always integer specializations, i.e., the assertion of the theorem holds even if we demand 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 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 K^alg, where K^alg is the algebraic closure of K.

  Applications

• The inverse Galois problem, Hilbert's original motivation. The theorem almost immediately implies that if a finite group G can be realized as the Galois group of a Galois extension N of
• Construction of elliptic curves with large rank.
• Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's last theorem.
• If a polynomial is a perfect square for all large integer values of x, then g is the square of a polynomial in This follows from Hilbert's irreducibility theorem with and

  Generalizations