Paper 4, Section II, H

Number Fields
Part II, 2017

(a) Write down OK\mathcal{O}_{K}, when K=Q(δ)K=\mathbb{Q}(\sqrt{\delta}), and δ2\delta \equiv 2 or 3(mod4)3(\bmod 4). [You need not prove your answer.]

Let L=Q(2,δ)L=\mathbb{Q}(\sqrt{2}, \sqrt{\delta}), where δ3(mod4)\delta \equiv 3(\bmod 4) is a square-free integer. Find an integral basis of OL\mathcal{O}_{L} \cdot [Hint: Begin by considering the relative traces trL/Kt r_{L / K}, for KK a quadratic subfield of L.]L .]

(b) Compute the ideal class group of Q(14)\mathbb{Q}(\sqrt{-14}).