monic degree-n polynomial fx = x n + fn−1x n−1 + · · · + f0 ∈ Zx that is irreducible modulo every prime p dividing q

Such a polynomial is called a "primitive polynomial" of degree n over Z. It is important to note that not every irreducible polynomial over Z is primitive.

Primitive polynomials have applications in various fields, including coding theory, cryptography, and number theory. In coding theory, primitive polynomials are used to generate finite fields, which are important for error-correcting codes. In cryptography, they are used in the construction of pseudorandom number generators. In number theory, they are used to construct primitive roots modulo a prime.

Finding primitive polynomials of a given degree can be a challenging problem. However, there are algorithms that can efficiently generate them. One such algorithm is the Zassenhaus algorithm, which is based on the factorization of cyclotomic polynomials. Another algorithm is the Lidl-Niederreiter algorithm, which is based on the construction of irreducible polynomials over finite fields.