MA3412S2_Hil2014.pdf

# If the multiplier c is neither a unit nor a prime

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If the multiplier c is neither a unit nor a prime element of R then it is the product of a finite number of prime elements of R , because R is a unique factorization domain. We have proved the result in the special case where the multiplier is a prime element of R . It follows that if the primitive polynomial f ( x ) divides p 1 p 2 · · · p k g ( x ) then f ( x ) divides p 2 p 3 · · · p k g ( x ). The result in the general case therefore follows by induction on the number of prime factors of the multiplier. Let R be a unique factorization domain. The units of the polynomial ring R [ x ] are the polynomials of degree zero whose coeffficients are units of the ring R . (Thus a polynomial with coefficients in R is a unit of R [ x ] if and only if it is a ‘constant polynomial’ whose ‘value’ is a unit of R .) It is not possible for a polynomial of degree zero to divide a primitive polynomial unless it is a unit of R [ x ]. It follows that a primitive polynomial 32

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with coefficients in R is an irreducible element of R [ x ] if and only if it cannot be factored as a product of polynomials of lower degree with coefficients in R . We define an irreducible primitive polynomial with coefficients in R to be a primitive polynomial of degree greater than zero that cannot be factored as a product of polynomials of lower degree. Lemma 2.29 Let R be a unique factorization domain. Then the irreducible elements of the polynomial ring R are the polynomials of degree zero whose coefficients are prime elements of R and the irreducible primitive polynomials of positive degree. Moreover every non-zero polynomial in R [ x ] that is not itself a unit of R [ x ] may be factored as a product of one or more irreducible elements of R [ x ] . Proof The subring of R [ x ] consisting of the polynomials of degree zero is isomorphic to the coefficient ring R , and the factors of a polynomial of degree zero must themselves be polynomials of degree zero. It follows that the irreducible elements of R [ x ] that are of degree zero are those polynomials of degree zero whose coefficients are prime elements of R [ x ]. Any polynomial of positive degree that is not primitive is divisible by some non-zero element of the coefficient ring R that is not a unit of R , and thus cannot be an irreducible element of R . It follows that the irreducible elements of R [ x ] that are of positive degree are the irreducible primitive polynomials with coefficients in R . Any primitive polynomial of positive degree with coefficients in R that is not itself an irreducible primitive polynomial can be factored as a product of polynomials of lower degree. The factors must themselves be primitive polynomials. It follows by induction on the degree of the primitive poly- nomial that any primitive polynomial of positive degree with coefficients in R can be factored as a product of a finite number of irreducible primitive polynomials. Therefore any non-zero polynomial with coefficients in R that is not a unit of R [ x ] can be factored as the product of a polynomial of degree
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