Solvability by Radicals

Suppose \(f(x) \in \mathbb {Q}[x]\) is a polynomial with Galois group of order \(594 = 2 \cdot 3^3 \cdot 11\). Show that \(f\) is solvable in radicals.

Let \(f(x)\) be a degree 5 polynomial in \(\mathbb {Q}[x]\) that is not solvable by radicals. Let \(L\) be its splitting field over \(\mathbb {Q}\).

1. Prove that there is at most one field \(K\) with \(\mathbb {Q}\subset K\subset L\) and \([K:\mathbb {Q}]=2\).
2. If \(\alpha ,\beta \) are irrational elements in \(L\) such that \(\alpha ^2\) and \(\beta ^2\) are rational, prove that \(\alpha \beta \) is rational.