The rational numbers, and every number field, are Hilbertian fields. The classical integral form supplies infinitely many integral specializations avoiding a prescribed finite exceptional set. The theorem preserves irreducibility, rather than asserting that every specialization works: becomes reducible at square parameters.
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Hilbert's irreducibility theorem is a result in algebraic number theory, specifically related to the behavior of certain types of polynomial equations. Formulated by David Hilbert in the early 20th century, the theorem provides a significant insight into the irreducibility of polynomials over number fields.