Download Mathematical Tripos Part III Paper 24 - Local Fields and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Tuesday, 7 June, 2011 9:00 am to 11:00 am PAPER 24 LOCAL FIELDS Attempt no more than THREE questions. There are FIVE questions in total. The questions carry equal weight. 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. 2 1 (a) Let k be a finite field. Define the field of Laurent power series K = k((t)). Show that there is a non-archimedean absolute value | · | on K such that (i) R = k[[t]] is the completion of k[t], (ii) K = k((t)) is the completion of k(t), (iii) #(R/fR) = |f |−1 for all 0 6= f ∈ R. (b) Let K be a field complete with respect to a discrete valuation. Show that K is locally compact if and only if it has finite residue field. 2 Let K be a finite Galois extension of Qp of degree n. (a) Show that if non-trivial absolute values | · |1 and | · |2 on K induce the same topology then there exists c > 0 such that |x|1 = |x|c2 for all x ∈ K. (b) Show that if | · | is an absolute value on K extending the p-adic absolute value | · |p on Qp then |x| = |NK/Qp(x)|1/np for all x ∈ K. [You should prove any results you need about equivalence of norms on vector spaces.] 3 (a) State and prove a version of Hensel’s Lemma. (b) Determine the number of solutions of x3 − 11x+ 40 = 0 in Zp for p = 2, 3, 5. (c) Let L ⊃ K be finite extensions of Qp. Show that if L/K is unramified then L/K is Galois. 4 Let K be a finite extension of Qp with valuation ring OK and residue field k of order q. Assuming any properties you need of the Teichmüller map, prove that (a) O∗K contains a subgroup of finite index isomorphic to (OK ,+). (b) If e(K/Qp) < p− 1 then K contains exactly q − 1 roots of unity. (c) Q∗p/(Q ∗ p) 3∼=Z/3Z or (Z/3Z)2. Part III, Paper 24
