Let $K$ be a number field. State Dirichlet's unit theorem, defining all the terms you use, and what it implies for a quadratic field $\mathbb{Q}(\sqrt{d})$, where $d \neq 0,1$ is a square-free integer.

Find a fundamental unit of $\mathbb{Q}(\sqrt{26})$.

Find all integral solutions $(x, y)$ of the equation $x^{2}-26 y^{2}=\pm 10$.