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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6320 • Algebra II

Fall 2022 • Question 5

Question

Math 6320 • Algebra II Fall 2022 • Question 5

Let \(f(x)\) be an irreducible polynomial in \(\mathbb {Q}[x]\), with splitting field \(K\). Suppose \(\operatorname {Gal}(K/\mathbb {Q})\) is abelian. Prove or disprove: \(K = \mathbb {Q}(\alpha )\) for each root \(\alpha \) of \(f(x)\) in \(K\).

Raw LaTeX

Let \(f(x)\) be an irreducible polynomial in \(\mathbb{Q}[x]\), with splitting
field \(K\). Suppose \(\operatorname{Gal}(K/\mathbb{Q})\) is abelian. Prove or
disprove: \(K = \mathbb{Q}(\alpha)\) for each root \(\alpha\) of \(f(x)\) in \(K\).

Topic Tags

Field Extensions Galois Groups of Polynomials Irreducible Polynomials Splitting Fields