Cyclotomic Fields - Mathematical Tripos - Past Exam, Exams of Mathematics

Download Cyclotomic Fields - Mathematical Tripos - Past Exam and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Tuesday 14 June, 2005 9 to 12 PAPER 30 CYCLOTOMIC FIELDS Attempt ALL questions. There are FOUR questions in total. The questions carry equal weight. Notation: Throughout, p will denote an odd prime, and Zp the ring of p-adic integers. For any ring A, A∗ will denote the multiplicative group of units of A. Let R = Zp [[T ]] be the ring of formal power series in an indeterminate T with coefficients in Zp. Finally, φ : R → R will be the ring homomorphism defined by φ(f)(T ) = f((1 + T )p − 1). STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.