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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2008 • Question 10

Question

Math 6310 • Algebra Fall 2008 • Question 10

Let \(\omega =e^{2\pi i/3}\). Let \(\sigma \) and \(\tau \) be automorphisms of \(\mathbb {C}(x)\) which fix \(\mathbb {C}\) and satisfy \[ \sigma (x)=\omega x,\qquad \tau (x)=1/x. \] Prove that the group \(\langle \sigma ,\tau \rangle \) is isomorphic to the symmetric group \(S_3\), and determine the subfield of \(\mathbb {C}(x)\) that is fixed by this group.

Raw LaTeX

Let \(\omega=e^{2\pi i/3}\). Let \(\sigma\) and \(\tau\) be automorphisms of \(\mathbb{C}(x)\) which
fix \(\mathbb{C}\) and satisfy
\[
\sigma(x)=\omega x,\qquad \tau(x)=1/x.
\]
Prove that the group \(\langle\sigma,\tau\rangle\) is isomorphic to the symmetric group \(S_3\), and
determine the subfield of \(\mathbb{C}(x)\) that is fixed by this group.

Topic Tags

Cyclotomic Fields Field Extensions Symmetric and Alternating Groups