  The Schinzel hypothesis is a famous conjectural statement about primes in
value sets of polynomials, which generalizes the Dirichlet theorem about primes
in an arithmetic progression. We consider the situation that the ring of
integers is replaced by a polynomial ring and prove the Schinzel hypothesis for
a wide class of them: polynomials in at least one variable over the integers,
polynomials in several variables over an arbitrary field, etc. We achieve this
goal by developing a version over rings of the Hilbert specialization property.
A polynomial Goldbach conjecture is deduced, along with a result on spectra of
rational functions.
