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(a) If R is a unique factorization domain, prove that every f(x) belongs to R[x] can be written as f(x) = af_1(x) where a belongs to R and where f_1(x) is primitive, (b) Prove that the decomposition in part (a) is unique ( up to associate).

Modern Algebra
Ring Theory (XLVIII)
Unique Factorization Domain
Greatest Common Divisor
Relatively Prime Elements
Integral Domain
Irreducible Elements
Unit of a Commutative Ring
Associates

(a) If R is a unique factorization domain, prove that every f(x) belongs to R[x] can be written
as f(x) = af_1(x) where a belongs to R and where f_1(x) is primitive.

(b) Prove that the decomposition in part (a) is unique (up to associate).

See attached file for full problem description.

Solution Summary

This solution is comprised of a detailed explanation of the properties of polynomials in R[x],
where R is a unique factorization domain.
It contains step-by-step explanation to prove that
(a) if R is a unique factorization domain, then every f(x) belongs to R[x] can be written
as f(x) = af_1(x) where a belongs to R and where f_1(x) is primitive and
(b) Prove that the decomposition in part (a) is unique (up to associate).

Notes are also given at the end.

Solution contains detailed step-by-step explanation.

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