(a) Explain what is meant by an integral basis of an algebraic number field. Specify such a basis for the quadratic field $k=\mathbb{Q}(\sqrt{2})$.

(b) Let $K=\mathbb{Q}(\alpha)$ with $\alpha=\sqrt[4]{2}$, a fourth root of 2 . Write an element $\theta$ of $K$ as

$$\theta=a+b \alpha+c \alpha^{2}+d \alpha^{3}$$

with $a, b, c, d \in \mathbb{Q}$. By computing the relative traces $T_{K / k}(\theta)$ and $T_{K / k}(\alpha \theta)$, show that if $\theta$ is an algebraic integer of $K$, then $2 a, 2 b, 2 c$ and $4 d$ are rational integers. By further computing the relative norm $N_{K / k}(\theta)$, show that

$$a^{2}+2 c^{2}-4 b d \text { and } 2 a c-b^{2}-2 d^{2}$$

are rational integers. Deduce that $1, \alpha, \alpha^{2}, \alpha^{3}$ is an integral basis of $K$.