Exercise 8, § 1

Let K be a quadratic field, that is an extension of $\mathbf{Q}$ of degree 2. a) Show that $K = \mathbf{Q}(e)$ where $e^2$ is a rational integer $d \neq 1$ not divisible by a square $> 1$ (in $Z$). b) If $a = a + be$ ($a, b \in \mathbf{Q}$), let $\bar{\alpha}$ denote the conjugate of $a + be$. We say that $a$ is an integer of the quadratic field K if its norm $\alpha \bar{\alpha}$ and its trace $a + \bar{\alpha}$ are rational integers. Show that the integers of K form a ring A (notice that if $\alpha$ and $\beta$ are integers of K then the sum and product of the rational numbers $\alpha \beta + \overline{\alpha \beta}$ and $\alpha \overline{\beta} + \overline{\alpha \beta}$ are integers, hence that these rational numbers are the roots of a monic polynomial over $Z$; deduce that they are rational integers). c) Show that the integers of $K = \mathbf{Q}(e)$ form a $Z$-module with basis $(1, e)$ if $d - 1 \not\equiv 0 \pmod{4}$, or $(1/2(1 + e), 1/2(1 - e))$ if $d - 1 \equiv 0 \pmod{4}$ (if $a + be$ is an integer of K, check that $2a$ and $2b$ are rational integers such that $(2a)^2 - d(2b)^2 \equiv 0(4)$, and investigate whether they can be odd). Show that the discriminants of these bases (III, p. 549, Def. 3) are $4d$ and $d$ respectively.