Let d be a square-free integer.

(i) Show that every element of the ring Z[d] is a root of a polynomial of the form x^2 +ax + b Z[x].

(ii) Assume that d different than 1 (mod 4). Let , Q. Show that if + d is a root of a polynomial of the form x^2 + ax + b Z[x] then + d Z[d].

(iii) By exhibiting a counterexample for a specific d show that (ii) may be false if d 1

(mod 4).