1. An irreducible element in is either an irreducible element in or an irreducible primitive polynomial . 2. If is not, it is a primitive polynomial ( because it is irreducible ). 3. A consequence is that factoring polynomials over the rationals is equivalent to factoring primitive polynomials over the integers. 4. This defines a factorization of " p " into the product of an integer and a primitive polynomial . 5. That lemma says that if the polynomial factors in, then it also factors in as a product of primitive polynomials . 6. This implies that a primitive polynomial is irreducible over the rationals if and only if it is irreducible over the integers. 7. If " R " is a GCD domain, then the set of primitive polynomials in is closed under multiplication. 8. Over a unique factorization domain the same theorem is true, but is more accurately formulated by using the notion of primitive polynomial . 9. A primitive polynomial is a polynomial over a unique factorization domain, such that 1 is a greatest common divisor of its coefficients. 10. A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by " x ".
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