Chapter 14

Suppose that \(G\) is a transitive subgroup of \(\Sigma_{4}\). Show that \(G\) is either (i) \(\Sigma_{4}\), (ii) \(A_{4}\), (iii) the Viergruppe \(N\), (iv) cyclic of order 4 or \((v)\) a nonabelian group of order 8, isomorphic to the group of rotations and reflections of a square.

Determine the Galois groups of the following quartics in \(\mathbb{Q}[x]:\) (i) \(x^{4}+4 x+2\) (ii) \(x^{4}+8 x-12\) (iii) \(x^{4}+1\); (iv) \(x^{4}+x^{3}+x^{2}+x+1\) (v) \(x^{4}-2\)