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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2019 • Question 10

Question

Math 6310 • Algebra Spring 2019 • Question 10

Let \(K\) be a field, and let \(L\) be the splitting field of an irreducible and separable polynomial \(g(x)\in K[x]\). If \(\operatorname {Gal}(L/K)\) is abelian, and \(\alpha \in L\) is a root of \(g(x)\), prove that \(L=K(\alpha )\).

Raw LaTeX

Let \(K\) be a field, and let \(L\) be the splitting field of an irreducible and separable polynomial
\(g(x)\in K[x]\). If \(\operatorname{Gal}(L/K)\) is abelian, and \(\alpha\in L\) is a root of \(g(x)\),
prove that \(L=K(\alpha)\).

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Irreducible Polynomials Polynomial Rings and Ring Homomorphism Kernels Splitting Fields