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I think I can construct an explicit counterexample with $a\in\mathbb{Q}_p$. Choose a compatible sequence $\zeta_{p^m}$ of $p^m$th roots of unity in $\overline{\mathbb{Q}}_p$. Write $q=p^n$ with $n\g …

Oh, I know how I would try and build examples. First I would write down a random $A_5$ extension $K$ of $\mathbf{Q}$, ramified at some primes (in fact I would look in a table, e.g. in Buhler's thesis …

Fields with the same Galois closure can be completely different. For example take a random irreducible polynomial of huge degree, let $F$ be the field obtained by adjoining one root of the polynomial, …

This answer is just to bump this post up to the front page for the final time. I typed up all the errata I heard into one pdf file and put it here. The London Maths Society would like comments, if any …

EDIT: Hendrik Lenstra emailed me a proof of Conjecture 2. I'll append it below. So Jagy's question is now solved. OK so I think that Jagy wants to make the following conjecture: CONJECTURE 1: an i …

Brian Conrad says: Page 52, section 8, Definition, displayed expression: put subscripts outside norms. page 53, 2 lines above section 9: a_N rather than a_n in first norm on right side Page 56, The …

Hendrik Lenstra says: Below my 51 errata that I didn't see on your list or in William Stein's mail yet. Most are of a typographical nature, but some have mathematical substance. I did at the present …

Rebecca Bellovin writes: Here's one I didn't see on the list on mathoverflow: In exercise 2, part 10, equation (**) (page 353, line 4) should be $$\prs{\lambda}{b}=\prod_{v\in S}(b,\lambda)_{v} …

Keith Conrad sent me a nice chunky list here.

OK how's about this to finish (I don't think either argument posted so far deals with this case). Say $K/\mathbf{Q}$ is finite and (away from a finite set of exceptions) $p$ splits completely in $K$ i …

Joseph Oesterlé says: p 69, l 26 S contains all v with |alpha_v|_v < 1 should be S contains all v with |alpha_v|_v ≠ 1 p 69, l 27 \frac{1}{2}C should be \frac{1}{2C} p 131 corollary 2. "Let L/K be …

Dominique Bernardi points out that the formula for $\phi_a$ is wrong on line 1 of p96. This is a delicate one. The issue is what the definition of the action of $G$ on $A*$ is (NB that starshould be a …

Eric Bach writes to tell me that there should arguably be a +- sign in front of the formula for the discriminant of the $m$th cyclotomic field appearing on pp88--89 (or perhaps it should be flagged in …

I posted this question in several other places as well (the nmbrthry mailing list, and sci.math.research). Here, completely unedited, is the bulk of an email I just got from Rene Schoof. page 30 lin …

And here's one which I spotted: I think that the last full sentence at the bottom of p98 is wrong. I think the "action" they define is not an action, and I think the first couple of sentences of secti …