From dinakar@its.caltech.edu Thu Oct 7 22:17:24 1999 Return-Path: Received: from blinky.its.caltech.edu (blinky.its.caltech.edu [131.215.48.132]) by zorn2.math.leidenuniv.nl (8.9.3/8.8.7) with ESMTP id WAA30264 for ; Thu, 7 Oct 1999 22:17:23 +0200 Received: from localhost (dinakar@localhost) by blinky.its.caltech.edu (8.9.1/8.9.1) with ESMTP id NAA25132; Thu, 7 Oct 1999 13:15:01 -0700 (PDT) X-Authentication-Warning: blinky.its.caltech.edu: dinakar owned process doing -bs Date: Thu, 7 Oct 1999 13:15:01 -0700 (PDT) From: Dinakar Ramakrishnan X-Sender: dinakar@blinky To: Peter Stevenhagen cc: desmit@math.leidenuniv.nl, drama@ias.edu Subject: Re: FANF In-Reply-To: <199910071610.SAA13244@noot.wins.uva.nl> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Dear Peter and Bart (if I may), (you may address me as Dinakar; we are not so formal in the States) Thanks for your message. I am on sabbatical this year at the IAS. > > You may be pleased to hear that we have selected your > book `Fourier analysis on number fields' as a text > for our local graduate student seminar in number theory. > Thanks. > We are a bit confused by the conventions on topological groups > that you introduce in the first chapter. > Topological groups are not assumed to be T_1 (page 5), so for > all we know the topology might be indiscrete. But how can we > then prove that the closure of an abelian subgroup (such as {e}) > is always abelian (exercise 1.7)? You are right. We neglected to tell the reader to assume that the group is Hausdorff for this exercise. > Even more troubling, you prove in theorem 1.18 (page 31) that > subgroups of finite index in a profinite group are always open. > If we take G to be a countably infinite product of fields F2 > (of two elements), then G is a topological ring. Any maximal ideal > containing the direct sum of all F2's inside the direct product G > is an additive subgroup of index 2 that is not open in G. > Again you are right. I should have proofread carefully as I am aware of this example and the pitfalls. The statement of Thm.1-18 should say that a closed subgroup H of a profinite group G is open iff G/H is finite. The proof as given seems to go through with this correction. I have two comments: (i) As I learnt from Serre a while ago, one can prove the following: If H is a topologically finitely generated subgroup of a profinite group G with G/H finite, then H is open. (ii) I will be very interested if you can come up with a counterexample which does not use the axiom of choice. ________________________________________________________________________ It is certainly embarrassing to find such errors in one's book. But I am grateful to be apparaised of them so I can correct them in the next printing or issue a sheet of errata. Best. Dinakar