MA3412S2_Hil2014.pdf

# Zero and a primitive polynomial and can therefore be

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zero and a primitive polynomial, and can therefore be factored as a product of irreducible elements of the polynomial ring R [ x ]. Lemma 2.30 Let R be a unique factorization domain. Then any polynomial of degree zero whose coefficient is a prime element of R is a prime element of the polynomial ring R [ x ] . Proof Let p be a prime element of R , and let g ( x ) and h ( x ) be polynomials with coefficients in R . Then there exist primitive polynomials ˆ g ( x ) and ˆ h ( x ) and elements a and b of R such that g ( x ) = a ˆ g ( x ) and h ( x ) = b ˆ h ( x ). Suppose 33

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that p divides all the coefficients of the polynomial g ( x ) h ( x ). Now g ( x ) h ( x ) = ab ˆ g ( x ) ˆ h ( x ). Moroever ˆ g ( x ) ˆ h ( x ) is a primitive polynomial, because the prod- uct of two primitive polynomials is always primitive (Lemma 2.27). It follows that there must exist at least one coefficient of ˆ g ( x ) ˆ h ( x ) that is not divisible by the prime element p of R , and therefore ab is divisible by p . But then either a is divisible by p , in which case all the coefficients of g ( x ) are divisible by p , or else b is divisible by p , in which case all the coefficients of h ( x ) are divisible by p . Thus the polynomial of degree zero with coefficient p is a prime element of the polynomial ring R [ x ]. Lemma 2.31 Let R be an integral domain, let I be a non-zero ideal of the polynomial ring R [ x ] , and let w ( x ) be a non-zero polynomial belonging to I whose degree is less than or equal to the degree of every other non-zero poly- nomial belonging to the ideal I . Then, given any polynomial g ( x ) belonging to I , there exists some non-zero element c of R with the property that cg ( x ) is divisible in R [ x ] by w [ x ] . Proof Let m be the degree of the polynomial w ( x ), and let let k be an integer satisfying k m . Suppose that, given any non-zero polynomial h ( x ) in I of degree less than k , there exists some non-zero element b of R with the property that bh ( x ) is divisible by w ( x ) in R [ x ]. Let g ( x ) be a polynomial in I of degree k . Then there exist non-zero elements d and a of R such that dg ( x ) and ax k - m w ( x ) have the same leading coefficient. Then either dg ( x ) - ax k - m w ( x ) = 0 R or else dg ( x ) - ax k - m w ( x ) is a non-zero polynomial belonging to the ideal I whose degree is less than k . There must then exist some non-zero element b for which b ( dg ( x ) - ax k - 1 w ( x )) is divisible by w ( x ). Let c = bd . Then c is non-zero and cg ( x ) is divisible in R [ x ] by w [ x ]. The result therefore follows by induction on the degree of the polynomial g ( x ). Lemma 2.32 Let R be a unique factorization domain, let f ( x ) be an irre- ducible primitive polynomial with coefficients in R , and let g ( x ) and h ( x ) be polynomials with coefficients in R . Suppose that f ( x ) divides g ( x ) h ( x ) in R [ x ] . Then either f ( x ) divides g ( x ) in R [ x ] or f ( x ) divides h ( x ) in R [ x ] . Thus every irreducible primitive polynomial with coefficients in R is a prime element of R [ x ] .
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