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Question

Math 6310 • Algebra

Spring 2017 • Question 8

Question

Math 6310 • Algebra Spring 2017 • Question 8

Let \(K=\mathbb {Q}(\pi )\), where \(\pi \) is the ratio of the circumference of a circle to its diameter. You may assume without proof that \(\pi \) is transcendental over \(\mathbb {Q}\).

  1. Show that \(f(x)=x^3-\pi \) is irreducible over \(K\).
  2. Let \(L\) denote the splitting field of \(f(x)\) over \(K\). Compute \([L:K]\) and \(\operatorname {Gal}(L/K)\).

Raw LaTeX

Let \(K=\mathbb{Q}(\pi)\), where \(\pi\) is the ratio of the circumference of a circle to its
diameter. You may assume without proof that \(\pi\) is transcendental over \(\mathbb{Q}\).

\begin{enumerate}
\item Show that \(f(x)=x^3-\pi\) is irreducible over \(K\).
\item Let \(L\) denote the splitting field of \(f(x)\) over \(K\). Compute \([L:K]\) and
\(\operatorname{Gal}(L/K)\).
\end{enumerate}

Topic Tags

Irreducible Polynomials Splitting Fields