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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2001 • Question 5

Question

Math 6310 • Algebra Fall 2001 • Question 5

Let \(R\) be a unique factorization domain with quotient field \(K\). Let \(f(X)\) be a polynomial in \(R[X]\). Show that if \(f(X)\) can be factored into polynomials of lower degree in \(K[X]\), then \(f(X)\) can be factored into polynomials of lower degree in \(R[X]\).

Raw LaTeX

Let \(R\) be a unique factorization domain with quotient field \(K\). Let \(f(X)\) be a polynomial in
\(R[X]\). Show that if \(f(X)\) can be factored into polynomials of lower degree in \(K[X]\), then
\(f(X)\) can be factored into polynomials of lower degree in \(R[X]\).

Topic Tags

Galois Groups of Polynomials Ideals and Quotient Rings Irreducible Polynomials Polynomial Rings and Ring Homomorphism Kernels