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