Exercise 11, § 1

We investigate the quadratic fields Q(\sqrt{d}) for which the map a \mapsto |N(α)| is a Euclidean function on the integers of the field (Ex. 7). a) Show that a necessary and sufficient condition for this is that, for all a \in Q(\sqrt{d}), there exist an integer β of the field such that |N(a - β)| < 1 ; write down this condition, expressing a and β in terms of an integer basis of the field (Ex. 8, c)). b) In the case of an imaginary quadratic field (d < 0), show that the only values of d for which |N(α)| is a Euclidean function are −1, −2, −3, −7, −11. c) For real quadratic fields, we will restrict ourselves to the cases d ≡ 2 or 3 (mod 4). Show that there are then only finitely many values of d such that |N(α)| is a Euclidean function. (If |N(α)| is a Euclidean function and a is a natural number such that 0 ≤ a ≤ d and a ≡ t^2(d), then show that either a or d − a has the form x^2 − dy^2, where x and y are rational integers, by considering the element t / \sqrt{d} of Q(\sqrt{d}) ; next show that for large enough d there exists an odd integer z such that 5 d < z^2 < 6d (if d ≡ 3 mod 4) or 2d < z^2 < 3d (if d ≡ 2 mod 4) ; then take a = z^2 − d[z^2/d] and show that, under these conditions, neither a nor d − a has the form x^2 − dy^2, by examining residues mod 8). Show that the only values of d for which no such integer z exists are 3, 7, 11, 19, 35, 47 and 59 (for d ≡ 3 mod 4) and 2, 6, 14 and 26 (for d ≡ 2 mod 4). Show that there is indeed a Euclidean function for d = 2 and d = 3.