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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2007 • Question 10

Question

Math 6310 • Algebra Spring 2007 • Question 10

Let \(R\) be a unique factorization domain with quotient field \(k\). Let \(f\) be an element of \(R[x]\). Show that if \(f\) can be factored as a product of lower-degree polynomials in \(k[x]\), then \(f\) can be factored as a product of lower-degree polynomials in \(R[x]\).

Raw LaTeX

Let \(R\) be a unique factorization domain with quotient field \(k\). Let \(f\) be an element of
\(R[x]\). Show that if \(f\) can be factored as a product of lower-degree polynomials in \(k[x]\),
then \(f\) can be factored as a product of lower-degree polynomials in \(R[x]\).

Topic Tags

Galois Groups of Polynomials Ideals and Quotient Rings Polynomial Rings and Ring Homomorphism Kernels Tensor Products of Modules