Mathematical Tripos Part III, Paper 27: Cyclotomic Fields, Exams of Mathematics

Download Mathematical Tripos Part III, Paper 27: Cyclotomic Fields and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Friday 4 June, 2004 1.30 to 4.30 PAPER 27 CYCLOTOMIC FIELDS Attempt FOUR questions. There are four questions in total. The questions carry equal weight. Notation: Throughout p will denote an odd prime number, Zp will denote the p- adic integers, and Qp the field of p-adic numbers. For each n > 0, ζn will denote a primitive pn+1-th root of unity, with the property that ζpn+1 = ζn for all n > 0. Finally, Kn will denote the field Qp(ζn). You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.