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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2009 • Question 8

Question

Math 6310 • Algebra Spring 2009 • Question 8

Suppose \(f(x)\in \mathbb {Q}[x]\) is an irreducible polynomial with complex roots \(\alpha _1,\ldots ,\alpha _n\). Prove that \(\alpha _i-\alpha _j\notin \mathbb {Q}\) for all \(i\ne j\).

Raw LaTeX

Suppose \(f(x)\in\mathbb{Q}[x]\) is an irreducible polynomial with complex roots
\(\alpha_1,\ldots,\alpha_n\). Prove that \(\alpha_i-\alpha_j\notin\mathbb{Q}\) for all \(i\ne j\).

Topic Tags

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