 Solution

An integral basis of a degree-$d$ number field is a $\mathbb{Z}$-basis of its ring of integers: each algebraic integer has a unique integer linear combination of those $d$ elements. For $k=\mathbb{Q}(\sqrt{2})$, the standard quadratic-integer criterion gives ${\mathcal{O}}_k=\mathbb{Z}[\sqrt{2}]$, since $2$ is squarefree and is not congruent to one modulo four. Thus