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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2013 • Question 9

Question

Math 6310 • Algebra Spring 2013 • Question 9

Let \(f(x)\in \mathbb {Q}[x]\) be a polynomial of degree \(n\ge 4\) and let \(K\) be a splitting field of \(f\) over \(\mathbb {Q}\). Suppose that \(\operatorname {Gal}(K/\mathbb {Q})\) is the symmetric group \(S_n\). If \(\alpha \in K\) is a root of \(f(x)\), show that \(\alpha ^n\notin \mathbb {Q}\).

Raw LaTeX

Let \(f(x)\in\mathbb{Q}[x]\) be a polynomial of degree \(n\ge 4\) and let \(K\) be a splitting field
of \(f\) over \(\mathbb{Q}\). Suppose that \(\operatorname{Gal}(K/\mathbb{Q})\) is the symmetric
group \(S_n\). If \(\alpha\in K\) is a root of \(f(x)\), show that \(\alpha^n\notin\mathbb{Q}\).

Topic Tags

Galois Groups of Polynomials Polynomial Rings and Ring Homomorphism Kernels Splitting Fields Symmetric and Alternating Groups