Let $K=\mathbb{Q}(\sqrt{35})$. By Dedekind's theorem, or otherwise, show that the ideal equations

$$2=[2, \omega]^{2}, \quad 5=[5, \omega]^{2}, \quad[\omega]=[2, \omega][5, \omega]$$

hold in $K$, where $\omega=5+\sqrt{35}$. Deduce that $K$ has class number 2 .

Verify that $1+\omega$ is the fundamental unit in $K$. Hence show that the complete solution in integers $x, y$ of the equation $x^{2}-35 y^{2}=-10$ is given by

$$x+\sqrt{35} y=\pm \omega(1+\omega)^{n} \quad(n=0, \pm 1, \pm 2, \ldots) .$$

Calculate the particular solution $x, y$ for $n=1$.

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