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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6320 • Algebra II

Summer 2022 • Question 4

Question

Math 6320 • Algebra II Summer 2022 • Question 4

Let \(F\) be a field of characteristic zero and let \(E/F\) be a finite Galois extension. Suppose that \(E = F[\alpha ]\). Suppose that there exists \(\sigma \in G = \operatorname {Gal}(E/F)\) with \(\sigma (\alpha )=-\alpha \). Prove that \(E \ne F[\alpha ^2]\).

Raw LaTeX

Let \(F\) be a field of characteristic zero and let \(E/F\) be a finite Galois
extension. Suppose that \(E = F[\alpha]\). Suppose that there exists
\(\sigma \in G = \operatorname{Gal}(E/F)\) with \(\sigma(\alpha)=-\alpha\).
Prove that \(E \ne F[\alpha^2]\).

Topic Tags

Field Extensions Finite Fields Galois Theory