Exercise 1, § 1

Let d be a rational integer not divisible by a square in $\mathbf{Z}$. Show that the elements of the field $K = \mathbf{Q}(\sqrt{d})$ which are integral over $\mathbf{Z}$ are of the form $a + b\sqrt{d}$ where $a \in \mathbf{Z},\ b \in \mathbf{Z}$ if $d - 1 \not\equiv 0 \pmod{4}$ and the elements $(a + b\sqrt{d})/2$ where $a \in \mathbf{Z},\ b \in \mathbf{Z}$, $a$ and $b$ both even or $a$ and $b$ both odd, if $d - 1 \equiv 0 \pmod{4}$ (cf. Algebra, Chapter VII, § 1, Exercise 8). Deduce that $A = \mathbf{Z}[\sqrt{5}]$ is not integrally closed and give an example of an element of $\mathbf{Q}[\sqrt{5}]$ which is integral over $A$ but whose minimal polynomial over the field of fractions of $A$ does not have coefficients in $A$.