Explain what is meant by an integral basis $\omega_{1}, \ldots, \omega_{n}$ of a number field $K$. Give an expression for the discriminant of $K$ in terms of the traces of the $\omega_{i} \omega_{j}$.

Let $K=\mathbb{Q}(i, \sqrt{2})$. By computing the traces $T_{K / k}(\theta)$, where $k$ runs through the three quadratic subfields of $K$, show that the algebraic integers $\theta$ in $K$ have the form $\frac{1}{2}(\alpha+\beta \sqrt{2})$, where $\alpha=a+i b$ and $\beta=c+i d$ are Gaussian integers. By further computing the norm $N_{K / k}(\theta)$, where $k=\mathbb{Q}(\sqrt{2})$, show that $a$ and $b$ are even and that $c \equiv d(\bmod 2)$. Hence prove that an integral basis for $K$ is $1, i, \sqrt{2}, \frac{1}{2}(1+i) \sqrt{2}$.

Calculate the discriminant of $K$.