text: If a set of expressions has this property in a ring P, then there exists a tuple of rational numbers that can be substituted into the expressions which will also have this property in P, by a theorem of David Hilbert.
full_question: If a set of expressions has this property in a ring P, then there exists a tuple of rational numbers that can be substituted into the expressions which will also have this property in P, by a theorem of David Hilbert. A function that has a positive leading coefficient, coprime coefficients, and this property will produce infinitely many primes by the Bunyakovsky conjecture. By Gauss's lemma, a function that has this property with respect to the integers will have it over the rationals as well. For a collection of coefficients, if there exists a number p such that p doesn't divide the leading coefficient but it divides the rest, and p squared does not divide the constant term, then the polynomial will have this property by Eisenstein's criterion. The quadratic polynomial generated by factoring will have this property by the fundamental theorem of algebra. For 10 points, name this property of a polynomial, such that it can't be nontrivially factored over a field.
first_sentence: If a set of expressions has this property in a ring P, then there exists a tuple of rational numbers that can be substituted into the expressions which will also have this property in P, by a theorem of David Hilbert.
raw_answer: polynomial irreduciblity [or word forms; do NOT accept "reducible" or its word forms.]