Schinzel's theorem

Schinzel's theorem states that every nonconstant irreducible polynomial with integer coefficients has a root modulo infinitely many primes.

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Reduction modulo a prime: Reduction modulo a prime maps integer coefficients to residue classes in the finite field with that prime number of elements. A root modulo a prime is a root of the reduced polynomial in this finite field.
Integer polynomial: An integer polynomial is a polynomial whose coefficients are integers. Schinzel's hypothesis concerns irreducible polynomials drawn from this coefficient ring.
Andrzej Schinzel: Andrzej Schinzel was a Polish number theorist known for work on Diophantine problems and prime values of polynomials. Schinzel formulated the conjectural framework from which this theorem is a consequence.
Schur's theorem: Schur's theorem states that a nonconstant integer polynomial has infinitely many prime divisors among its nonzero values. Schur guarantees prime divisors of values, while Schinzel guarantees prime moduli with roots.
Quadratic residue: A quadratic residue modulo a prime is a residue class that is the square of another class modulo that prime. For an irreducible quadratic, a root modulo a prime means its discriminant is a quadratic residue there.
Finite field: A finite field is a field containing finitely many elements, whose size is a prime power. Each prime in the theorem supplies a finite field in which the polynomial has a root.
Prime number: A prime number is an integer greater than one with no positive divisors other than one and itself. The theorem counts distinct prime moduli, rather than roots over one fixed field.
Wacław Sierpiński: Wacław Sierpiński was a Polish mathematician who made major contributions to number theory, set theory, and topology. Sierpiński collaborated with Schinzel on the conjectures behind the theorem.
Linnik's theorem: Linnik's theorem bounds the least prime in a reduced arithmetic progression by a fixed power of its common difference. Linnik studies primes in prescribed progressions, not primes at which a polynomial has a root.
Modular arithmetic: Modular arithmetic performs integer calculations using remainders after division by a fixed modulus. The theorem's roots are solutions in modular arithmetic for infinitely many prime moduli.