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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2014 • Question 9

Question

Math 6310 • Algebra Spring 2014 • Question 9

Let \(\mathbb {F}\) be a field with \(\mathbb {Q}\subset \mathbb {F}\subset \mathbb {C}\) such that \([\mathbb {F}:\mathbb {Q}]\) is odd.

  1. If \(\mathbb {F}/\mathbb {Q}\) is Galois, prove that \(\mathbb {F}\) is contained in \(\mathbb {R}\).
  2. Find an extension with \([\mathbb {F}:\mathbb {Q}]=3\) such that \(\mathbb {F}\) is not contained in \(\mathbb {R}\).

Raw LaTeX

Let \(\mathbb{F}\) be a field with \(\mathbb{Q}\subset\mathbb{F}\subset\mathbb{C}\) such that
\([\mathbb{F}:\mathbb{Q}]\) is odd.
\begin{enumerate}
\item If \(\mathbb{F}/\mathbb{Q}\) is Galois, prove that \(\mathbb{F}\) is contained in
\(\mathbb{R}\).
\item Find an extension with \([\mathbb{F}:\mathbb{Q}]=3\) such that \(\mathbb{F}\) is not
contained in \(\mathbb{R}\).
\end{enumerate}

Topic Tags

Ext Functors and Module Extensions Field Extensions Finite Fields Galois Theory