(a) Let $f(X) \in \mathbb{Q}[X]$ be an irreducible polynomial of degree $n, \theta \in \mathbb{C}$ a root of $f$, and $K=\mathbb{Q}(\theta)$. Show that $\operatorname{disc}(f)=\pm N_{K / \mathbb{Q}}\left(f^{\prime}(\theta)\right)$.

(b) Now suppose $f(X)=X^{n}+a X+b$. Write down the matrix representing multiplication by $f^{\prime}(\theta)$ with respect to the basis $1, \theta, \ldots, \theta^{n-1}$ for $K$. Hence show that

$$\operatorname{disc}(f)=\pm\left((1-n)^{n-1} a^{n}+n^{n} b^{n-1}\right)$$

(c) Suppose $f(X)=X^{4}+X+1$. Determine $\mathcal{O}_{K}$. [You may quote any standard result, as long as you state it clearly.]