Please do it as fast as possible

1. Let m, n be coprime, square-free integers, other than 1. Let K = Q(\\m, Vn). (a) Let a E K. Prove that a E OK if and only if NK/Q(Vm)(@) and TI K/Q(Vm)(a) are algebraic integers. [8 marks] (b) Suppose that m = 3 mod 4 and n = 2 mod 4. By considering the traces TIK/Q(Vm)(@), TI K/Q(Vn)(a) and TI K/Q(1/mn) (a), or otherwise, show that every a E Or has the form NI- (k+l Vm+rvn+symn) for some k, l, r, s E Z. Further, by considering the norm NK/Q(/m)(a), or otherwise, show that k = / = 0 mod 2 and r = s mod 2. [11 marks] (c) Suppose again that m = 3 mod 4 and n = 2 mod 4. From (b), or otherwise, deduce that 1 1, Vm, Vn, 2 ( Vn+ Vmn) ; is a basis of Ox over Z. [3 marks] (d) Assuming that m = 3 mod 4 and n = 2 mod 4, prove that the discriminant of the algebraic number field K is 64m n2. [9 marks]