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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2006 • Question 4

Question

Math 6310 • Algebra Spring 2006 • Question 4

Prove Gauss’ Lemma over the integers: if \(h(x)\in \mathbb {Z}[x]\) such that \(h(x)=f(x)g(x)\) for nonconstant polynomials \(f(x),g(x)\in \mathbb {Q}[x]\), prove that there exist rational numbers \(a\) and \(b\) so that \(F(x)=af(x)\) and \(G(x)=bg(x)\) are in \(\mathbb {Z}[x]\) and so that \(h(x)=F(x)G(x)\).

Raw LaTeX

Prove Gauss' Lemma over the integers: if \(h(x)\in\mathbb{Z}[x]\) such that
\(h(x)=f(x)g(x)\) for nonconstant polynomials \(f(x),g(x)\in\mathbb{Q}[x]\), prove that there
exist rational numbers \(a\) and \(b\) so that \(F(x)=af(x)\) and \(G(x)=bg(x)\) are in
\(\mathbb{Z}[x]\) and so that \(h(x)=F(x)G(x)\).

Topic Tags

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