Let $K=\mathbb{Q}(\sqrt{26})$ and let $\varepsilon=5+\sqrt{26}$. By Dedekind's theorem, or otherwise, show that the ideal equations

$$2=[2, \varepsilon+1]^{2}, \quad 5=[5, \varepsilon+1][5, \varepsilon-1], \quad[\varepsilon+1]=[2, \varepsilon+1][5, \varepsilon+1]$$

hold in $K$. Deduce that $K$ has class number 2 .

Show that $\varepsilon$ is the fundamental unit in $K$. Hence verify that all solutions in integers $x, y$ of the equation $x^{2}-26 y^{2}=\pm 10$ are given by

$$x+\sqrt{26} y=\pm \varepsilon^{n}(\varepsilon \pm 1) \quad(n=0, \pm 1, \pm 2, \ldots) .$$

[It may be assumed that the Minkowski constant for $K$ is $\frac{1}{2}$.]