Hilbert's irreducibility theorem [SEP] named after
Hilbert's irreducibility theorem
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory.
Formulation of the theorem.
Hilbert's irreducibility theorem. Let
be irreducible polynomials in the ring
Then there exists an "r"-tuple of rational numbers
K"["X","Y"], where "K" is the algebraic closure of "K".
Applications.
Hilbert's irreducibility theorem has numerous applications in number theory and algebra. For example:
- 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