(a) Define the ideal class group of an algebraic number field $K$. State a result involving the discriminant of $K$ that implies that the ideal class group is finite.

(b) Put $K=\mathbb{Q}(\omega)$, where $\omega=\frac{1}{2}(1+\sqrt{-23})$, and let $\mathcal{O}_{K}$ be the ring of integers of $K$. Show that $\mathcal{O}_{K}=\mathbb{Z}+\mathbb{Z} \omega$. Factorise the ideals [2] and [3] in $\mathcal{O}_{K}$ into prime ideals. Verify that the equation of ideals

$$[2, \omega][3, \omega]=[\omega]$$

holds. Hence prove that $K$ has class number 3 .