Chapter 20

Show that if \(G\) is a soluble transitive subgroup of \(\Sigma_{p}\) (where \(p\) is a prime) then every element of \(G\) other than the identity fixes at most one point.

Show that if \(f\) is an irreducible polynomial in \(Q[x]\) of odd prime degree \(p\) which is solvable by radicals than either all the roots of \(f\) are real or \(f\) has exactly one real root. Show that if \(p=4 k+3\) then the discriminant can be used to distinguish the two possibilities. What happens if \(p=4 k+1 ?\)

Suppose that \(f\) is an irreducible polynomial of prime degree \(p\) in \(K[x]\), and that char \(K \neq p\). Let \(L: K\) be a splitting field extension for \(f\). Show that \(f\) is solvable by radicals if and only if whenever \(\alpha\) and \(\beta\) are distinct roots of \(f\) then \(L=K(\alpha, \beta)\).

Suppose that \(f\) is a quintic in \(\mathbb{Q}[x]\) whose Galois group contains \(D_{10}\). Show that the ten elements \(\alpha_{i}+\alpha_{j}(1 \leqslant i