Exercise 10, § 1

Let d be a rational integer not divisible by a square $> 1$. In the quadratic field $K = \mathbf{Q}(\sqrt{d})$, we shall examine the units, that is to say the invertible elements of the ring A of integers of K (Ex. 8, b)). a) If $d < 0$ ("imaginary quadratic fields"), then show that the only units of K are 1 and $-1$, except for $d = -1$, when i and $-i$ are also units, and for $d = -3$, where the four numbers $1/2(\pm 1 \mp \sqrt{-3})$ are also units. b) Suppose from now on that $d > 0$ ("real quadratic fields"), and suppose that K is embedded in a maximal ordered extension of $\mathbf{Q}$ (VI, p. 25). \* for example in R. \* Show that the positive units form a subgroup U' of the group U of units of K, and that U is the direct product of U' and {−1, 1}. c) Show that the units of K are the integers with norm 1 or −1. d) Show that the group U' is nontrivial (let D denote the discriminant of the integer basis (Ex. 8, c)) ; show that the integers a of K such that N(α) ≤ D are partitioned into a finite number of congruence classes mod U ; choose an integer β, in each class ; if B is a rational number such that 0 < B < inf (|β_i|), show (Ex. 9) that there exists an integer β of K such that 0 < |β| ≤ B and N(β) ≤ D ; deduce that some ββ_i^{-1} is a unit η such that |η| < 1). e) Show that U' is isomorphic to Z (show that there exists a unit ε such that |ε| < 1 and |ε| is as great as possible with this property, using Ex. 9 and the fact that |ξ̄| = |ξ| : for a unit). f) Show that in Q(\sqrt{2}), the element \sqrt{2} - 1 is a generator of U', and has norm −1. g) Show that if d has a prime factor p of the form 4n + 1, then every generator of U' has norm 1. (Observe that −1 cannot be a square in the field F_p, using V, p. 93, Prop. 1, c)).