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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2013 • Question 8

Question

Math 6310 • Algebra Fall 2013 • Question 8

Let \(\alpha \) be a complex root of \(x^6+3\). Set \(K=\mathbb {Q}(\alpha )\).

  1. Prove that \(K\) contains a primitive 6-th root of unity.
  2. Compute the Galois group of \(K\) over \(\mathbb {Q}\).

Raw LaTeX

Let \(\alpha\) be a complex root of \(x^6+3\). Set \(K=\mathbb{Q}(\alpha)\).
\begin{enumerate}
\item
Prove that \(K\) contains a primitive 6-th root of unity.

\item
Compute the Galois group of \(K\) over \(\mathbb{Q}\).
\end{enumerate}

Topic Tags

Cyclotomic Fields Field Extensions Galois Theory