1.
Determine the minimal polynomial over \(\Q\) for the following elements.
\begin{equation*}
\sqrt{2}+\sqrt{5}\quad\text{and}\quad 1+\sqrt[3]{2}+\sqrt[3]{4}
\end{equation*}
Worksheet Tutorial-8
Following exercises are taken from Section 14.2 of the book Abstract Algebra by Dummit and Foote.
2.
Determine the Galois group of \((x^2-2)(x^2-3)(x^2-5)\) over \(\Q\text{.}\)Furthermore, determine all the subfields of the splitting field of this polynomial.
3.
Let \(p\) be a prime number. Determine the Galois group of \(x^p-2\) over \(\Q\) and over \(\F_p\text{.}\) Show that the Galois group over \(\F_p\) is isomorphic to
\begin{equation*}
\left\{\left(\begin{smallmatrix}a\amp b\\0\amp 1\end{smallmatrix}\right):a,b\in F_p\text{ and }a\neq 0\right\}.
\end{equation*}
4.
Let \(\Q\left(\sqrt[8]{2},i\right)\) and let \(F_1=\Q(i)\text{,}\) \(F_2=\Q\left(\sqrt{2}\right)\text{,}\) \(F_3=\Q\left(\sqrt{-2}\right)\text{.}\) Show that
\begin{equation*}
\Gal{K}{F_1}\simeq\Z/8\Z;\quad \Gal{K}{F_2}\simeq D_8;\quad \Gal{K}{F_3}\simeq Q_8.
\end{equation*}
5.
Determine all subfields of the splitting field of \(x^8-2\) which are Galois over \(\Q\text{.}\)
