Explain what is meant by an integral basis for a number field. Splitting into the cases $d \equiv 1(\bmod 4)$ and $d \equiv 2,3(\bmod 4)$, find an integral basis for $K=\mathbb{Q}(\sqrt{d})$ where $d \neq 0,1$ is a square-free integer. Justify your answer.

Find the fundamental unit in $\mathbb{Q}(\sqrt{13})$. Determine all integer solutions to the equation $x^{2}+x y-3 y^{2}=17$.