# Erratum for Cassels-Froehlich

Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).

IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has past, I've spent days going through the results, and I have collated them all here: Errata for Cassels-Froehlich. I will update this file as comments come in. The London Mathematical Society would like to know all the errors I've made myself, by 12th of February, so feel free to let me know of anything, however trivial!

EDIT: file updated 15th Feb.

Thanks to everyone who helped.

This one is very borderline and I certainly won't be offended/surprised if it gets closed. [EDIT: I clearly misjudged this---the question has a good few upvotes now.]

The London Maths Society (LMS) are thinking of (indeed, actively pursuing the idea of) reprinting "Algebraic number theory" Ed. Cassels and Froehlich. Hence the LMS had to go about contacting the authors of the original articles. When they contacted Serre he replied "sure reprint my articles, but please include the erratum that I indicated in my completed works." Sure enough, he had found a slip in one of his articles (in the statement of the "ugly lemma"---Serre went on to say that this had taught him not to abuse lemmas, as they might bite back!) and had taken the trouble to fix it when his completed works were published.

I looked over the thread from last October about the errata database but the database doesn't seem to contain this book. On the other hand it is surely a very widely-read book. I think I just found a typo in the definition of a co-induced module on p98: I think the $G$-action on $Hom_{G'}(\mathbf{Z}[G],A')$ ($G'$ a subgroup of $G$ and $A'$ a $G'$-module) should be that $g\in G$ sends $\phi$ to the function sending $g'$ to $\phi(g'g)$, not what the authors say (what they say isn't even an action as far as I can see, unless I made a slip). The notation is also terrible: $g'$ is in $G$, it seems to me.

Does anyone else have any scrawled marginal notes in their copies of Cassels-Froehlich about typos or other things that the LMS can fix? They are planning on having an erratum page at the beginning of the book when they reprint it.

EDIT: someone from the LMS got in touch with me to say that the only errata they would dare publish would be errata that had been confirmed by the authors, or someone "of a similar standing", so in fact it might be the case that not everything mentioned here gets put in the LMS erratum.

If this thread does get closed, feel free to email comments to buzzard at imperial.ac.uk.

Edit: Discussion on this thread is happening here at the meta.

Edit: the LMS have set a deadline of 1st Feb. After this point I (KB) will put all the errata we have caught into one file and the LMS will send it to the authors, asking for their approval.

Edit: Anton and Ilya have suggested that really this would be better if it had one big answer rather than lots of smaller ones. But let me persist with the "lots of smaller ones" for the time being, because I am still getting emails with non-trivial lists in from different sources and, although I want to put everything together into one pdf file, I don't really want to do it until I am pretty sure no more is coming in. On the other hand these partial lists have definitely been of help to some people, e.g. I've had emails saying "I have a big list of corrections; here are the ones that haven't already been mentioned."

Edit: OK, so all the people whom I was almost sure would have comments have now got back to me. I posted everything in one "burst" so as to only bump this post to the top one last time. What I will do now is to compile everything I have now (on the basis that I am not expecting much more) into one pdf list, and post a link to it.

Thanks to everyone that contributed.

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Why is this borderline? This actually seems to me an ideal use of the collected attention of Math Overflow to do something that helps the community more generally. – JSE Jan 11 '10 at 16:20
I agree, this seems like an excellent question. And I'm glad to hear that Cassels and Froelich may be reprinted! – David Speyer Jan 11 '10 at 16:22
Hmm. It's "borderline when it comes to what I want MO to be" ;-) (which is just loads of fun precisely-worded questions with precise answers). But judging by the upvotes I have misjudged this. I also emailed some people asking them if they had seen any typos. I'll post anything I get. – Kevin Buzzard Jan 11 '10 at 16:24
In my opinion this is an excellent example of a good community wiki question! – Grétar Amazeen Jan 11 '10 at 16:32
I thought of a use of upvotes/downvotes. If you think the typos really are typos, vote up :-) If you think it's reader error vote down! – Kevin Buzzard Jan 11 '10 at 22:06

Ok so it looks like I misjudged this and the community seem happy to have the question here, at least at present. So I figured I'd pass on the comments which Serre sent the LMS.

p.135, part b) of Lemma 4. Replace $H^q(H,M)$ by $H^q(K/H,M^H)$ and replace $\hat{H}^0(H,M)$ by $\hat{H}^0(K/H,M^H)$.

p.135, line 6 from bottom. Replace "to $G/H$" by "to $H$".

p.145. In Prop.6, replace "of Proposition 3" by "of Proposition 5".

p.225, line 11 (second line after Th.7). After "can be taken to be rational integers" add the following parenthesis : (provided one does not insist that the extensions $K/\Omega_{\ell}$ be cyclic).

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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 section 4 should be:

Let $G'$ be a subgroup of $G$. If $A'$ is a $G'$-module, we can form the $G$-module $A=Hom_{G'}(\Lambda,A')$. We give $A$ the following $G$-module structure: if $\phi\in A$,then $g.\phi$ is the homomorphism $h\mapsto\phi(hg)$. Then we have...

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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 line -16 we conclude that L_1 .... (rather than "the")

page 83 last line ...|f(alpha_0)|/|f'(alpha_0)| (index 0 missing)

page 86 line 16: numbering (one word)

page 86 line 17 Then f_1(x) .... (f is missing)

page 98 last line \phi(g'g) (no inverse) (think so)

page 99 line -17 The inflation map departs from H^q(G/H, A^H)

page 99 line -6 the map departs from H^q(G,A^t)

page 101 line 13 Hom(Z[G/H], A^*))

page 104 the top horizontal map in the commutative diagram should be delta^

page 106 formula (7.3) starts like (f.g)d= (df) ....

page 115 lin -3 follows from (ii) (rather than (iii))

page 113 Theorem 8 in the formulation one should add "for all p"

page 135 the ugly lemma ...you know already

page 141 Prop 2. Z should be bold face.

page 141 last lines of section 2.4 "q" is not the "q" of line 5 of section 2.4

page 144 line -4 I_\pi is topologically generated by sigma_pi

page 145 G = G_{L/K}

page 145 In Prop. 6 should be "(1) of prop 5 holds"

page 147 line -17 add "K is a local field"

page 148 line 7 f(X) = pX + .... (f is missing)

page 150 line -17 formula should be "f.phi^(p) - phi^(p).g... (rather than f)

page 153 line 14 ring A should be A^_nr (?)

page 155 line 5 of section 4.1 {Z \cup \infty} should be Z \cup {\infty}

page 157 line -2 f(chi) (f is missing)

page 173 section 5.6 "Number Field case" (rather than Number Theory Case)

page 177 last symbol on page should be J_L

page 195 in the big diagram the third top vertical arrow should not be there (at this point in the proof)

page 196 line 5 Im beta_1 \supset Im(inv_1)

page 242 in the diagram top left corner E_k (rather than E_K)

page 284 footnote: this is my favorite. After Stalin ... :-) [Note by KB: that's no typo!]

page 292 line -2 of the introduction "maximal unramified extension"

page 312 line -7 ....L_1(k^+ - 0) (bracket missing)

page 357 in displayed formula: f(tX + Y) ... (f missing)

EDIT: Serre says:

Schoof's list will be very useful; its length shows that the new edition will need at least one or two pages of corrections.

About that list: - p.106. It should be d(f.g) instead of (f.g)d because the coboundary of a cochain is written df and not fd.

• p.153. No correction is needed: the formal group is an A-module, not an A^_nr-module.

• p.195. I do not understand what change Schoof wants.

• p.284. In my copy of the book, the footnote does not refer to Stalin. The only mention of that name is on p.366; I seem to remember that it was Cassels who put it there, as a kind of inner joke.

One more typo: - p.88, l.17. The letter Z in Z[zeta] should be in boldface.

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Bill Stein independently got in touch with me to tell me about the two typos on p99 (he's giving a talk on that chapter in 30 minutes' time!) – Kevin Buzzard Jan 11 '10 at 23:10
I remember Serre explaining long ago that the reference to Stalin was a mischievous joke linked to NATO's sponsorship of the conference. – Georges Elencwajg Jan 12 '10 at 12:48
On p.140 Prop 1. line 7 of the proof, it is very confusing to write $\delta(r/n)\in\hat{H}^0(G,\mathbb{Z})$ and $\delta(r/n) =r$, since we know $\hat{H}^0(G,\mathbb{Z}) =\mathbb{Z}/n\mathbb{Z}$. I think this is a typo? Actually, I think the entire proof of Prop.1 is just restating the fact of line 6. It is tautological. – Ying Zhang Jan 10 '11 at 1:13

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, by Friday 12th Feb.

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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 some way that this is what the the "classical" discriminant is).

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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 superscript but the TeX interpreter is playing up on me). It is not explicitly defined in the paper, but the authors claim that $A*$ is co-induced on p96 and I think that this is hence an implicit definition. I figure that probably $A*=Hom(\Lambda,A)$ with the $G$-action on $A$ being thought of as trivial, so for $f\in A*$ we have $(\gamma.f)(\lambda)=f(\gamma^{-1}\lambda)$. If I got it right then the correct definition of $\phi_a$ is $\phi_a(g)=g^{-1}a$.

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Oesterle points out that the "standard" definition of a co-induced module is Hom(Lambda,X) with G-action (g.f)(lambda)=f(lambda.g). If this definition is used then what the authors write seems to be OK. However it seems to me that implicit in the article is a "non-standard" (but isomorphic to the standard) definition. – Kevin Buzzard Jan 12 '10 at 13:57

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 an extension" should be "Let L/K be a Galois extension"

p 133 proof of corollary 1 "Note that a less violent" should be "Note that if L/K is Galois, a less violent"

p 157 first line of section 4.2: One has to suppose L/K abelian.

p 209 l 11 to 19 Only the conclusion at the last line seems to be right...

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I happened to notice these misprints this morning :

p. 85--87 Replace $Q$ by $\mathbf{Q}$ at several places : p. 85, l. $-$4; p. 86, l. 7; p. 86, statement of Corollary 2; p. 87, l. 5; p. 87, l. $-$6 (twice); p. 87, l. $-$4; p. 87, l. $-$1.

p. 89 l.$-$5 Replace "automorph" by "automorphism".

p. 90 In the statement of Lemma 2, replace "($b$ must generate $K^\times/(K^\times)^n$)" by "($b$ must have order $n$ in $K^\times/(K^\times)^n$)".

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Keith Conrad sent me a nice chunky list here.

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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

    $$\prs{\lambda}{b}=\prod_{v\in S}(b,\lambda)_{v}$$


not

    $$\prs{\lambda}{b}=\prod_{v\in S}(\lambda,b)_{v}$$


(\prs = power residue symbol). The right-hand side should be the reciprocal of what it is in the book.

I recall also having trouble getting the signs in part 14 of that exercise (on cubic reciprocity) to work out, perhaps because of the mess of algebra, perhaps because some of the computations depend on part 10. I'm suspicious of equation (**); I'll try to check it again.

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I've started reading chapter 1. Could someone confirm that the following are typos?

Page 11, in the proof of proposition 1, $D_R(M)_p$ should be $D_R(M_p)$.

Page 18, start of section 5, $k$ should be $k_1$.

Page 18, in the proof of proposition 2, $f(pR_p/\bar{p})$ should be $f(pR_p/p)$.

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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 occasion not verify the correctness of those.

And: I did not do any proofreading of my list either!! I trust you will apply your own sound judgment.

Good luck!!

And best regards,

## Hendrik

Errata for Cassels-Fr\"ohlich copied by Hendrik Lenstra from his own copy Jan 13, 2010

Page 3, Proposition 1. This Proposition is misstated, and the proof has the wrong reference: Chapter II, section 10 has no such result, but Chapter II, section 5 does. The Theorem in the latter section is the correct formulation: it is not the extension of the valuation, but the completion that one wants to be unique. More or less coincidentally, the Proposition is correct as stated (exercise!), but that statement is neither used (by anybody) nor proved (in the book).

Page 45, line 5: for "=n" read "=n+1".

Page 52, part (3) of the first definition: for "K" read "V".

Page 54, line -5: replace roman "A" by italic "A" (twice).

Page 73, line 6: replace "vica" by "vice".

Page 75, line 1: replace "(19.9)" by "(19.10)".

Page 78, first line of display (A.19): replace "b_{ij}" by "b_{1j}".

Page 78, line -8 (display (A.24)): replace the third subscript "P" by "R".

Page 78, line -6: replace the third subscript "P" by "R".

Page 79, line -8: remove the three commas within the parentheses.

Page 98, the lower "delta" in the diagram should have a "hat" (the upper one has one, though it is barely visible in my copy).

Page 123, last line before section 2.5: replace roman "C" by italic "C".

Page 129, line 10: replace "]" by "])".

Page 130, line 1: replace "2.7" by "2.8".

Page 130, line 14: replace "2.5" by "1.5".

Page 131, last line before Corollary 1: replace "2.7" by "2.8".

Page 131, line -10: replace the second "K_{nr}" by "K_{nr}^*".

Page 135, line 6 of Lemma 4: replace the second "M)" by "M))".

Page 139, line 14 (the first display): put ")" before the final ".".

Page 140, line 3 of section 2.3: replace "H^2(G,Z)" by "H^1(G,Q/Z)" (with Q, Z boldface), since the isomorphism \delta hasn't been applied yet!

Page 140, line -8: replace "s" by "s_\alpha".

Page 140, line -2: replace "Prop. 2" by "Prop. 1". Also, the proof is confused. One defines s'\alpha to be (\alpha,L'/K), and the fact that s\alpha maps to s'_\alpha under the natural map G^{ab} -> (G/H)^{ab} FOLLOWS from the equality of character values rather than playing a role in the proof of that equality.

Page 141, first line after the first diagram: replace the last "K" by "K'".

Page 143, line -3: replace "Lubin" by "Lubin-".

Page 147, first line after Definition: put ")" at the end.

Page 150, proof of Proposition 1: (c) is not proved that way.

Page 150, line -10: for "left-and" read "left- and".

Page 151, line 13: replace the last "[a]" by "[b]".

Page 154, line 18: replace "r_\pi" by "r_\pi(\omega)".

Page 154, line 19: in my copy, there is the scrawled complaint "why is K_\pi from sec. 3.6 equal to K_\pi from section 2.8?", and a three line additional argument, which reads as follows: "Adopt the definition of K_\pi as in sec.3.6 (or Theorem 3(b)). Then r(\pi) is trivial on K_\pi (def. of r), and so is \vartheta(\pi) by Cor. to Prop. 6. Also r(\pi) and \vartheta(\pi) are F on K_{nr}. Hence r and \vartheta agree on \pi, hence on all of K^*. (Hence also K_\pi=K\pi !)

Page 154, line 2 of section 3.8: replace "2.3" by "2.7".

Page 154, line -8: replace "I_K" by "I'_K".

Page 155, line -11: replace roman "G" by italic "G".

Page 156, line 3: replace "3.3" by "3.4".

Page 156, line 10: replace "\beta_j" by "\beta^j".

Page 157, line 9: replace "intertia" by "inertia".

Page 158, line -4: replace "|" by "/".

Page 168, line 5: replace roman "F" by italic "F".

Page 168, line -16: replace roman "C" by italic "C".

Page 170, line -18: replace "U^S an arbitrarily small" by "U^S contained in an arbitrarily small" (because U^S is generally not open).

Page 175, line 2 after the diagram: for "N_{M/K}" read "N_{M/K}J_M".

Page 179, line 12: put ")" before the second "=".

Page 183, line 1: there is no "Proposition 2". Probably "Proposition 2.3" is meant.

Page 183, display (7): replace the second "K" by "K^*".

Page 192, line -11: replace "infiinte" by "infinite".

Page 211, line -12 (counting the footnote as -1): for "does or does not" read "does not or does". (This is what I scrawled, I did not verify it at the present occasion!)

Page 211, line -7: for "seq" read "seq.".

Page 236, line 5: for "2.5", read "1.2, Prop. 1".

Page 353, line 4: for "(\lambda,b)_v", read "(b,\lambda)_v".

Page 360, last line of Exercise 5.1: for "4.3", read "4.4".

Page 366, under "Tchebotarev, N.,", also list "165," and "227,".

----------------------THE---END-------------------------------------------------

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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, Theorem, line 1: "valuation | |" rather than "valuation $k$"

page 64, 2nd footnote at bottom: this seems wrong, since I can conceive of another topology: convergence on compacts using finite sets of normalized valuations on compacts (so to speak).

Page 70, line 12: product over v in S, not all v.

page 90, Lemma 2 parenthetical: b must have order n in K*/(K*)^n, not that it generates this quotient.

p. 91, line 7: \nu is over the wrong arrow

p. 92, line -6: factorization is in o_p[X], not o[X]

p. 92, line -2: g*_j, not f*_j.

p. 108, lines 3 and 4 of section 8: either make d begin at K_i or swap parities of i later in the sentence (I think).

p. 118, line -1: this "family" is a bit misleading, in that it is directed by U, not U \cap H.

p. 119, Corollary 3: again, indexing is unclear. Should be indexed by U in the limit process, not UH. Make it clearer.

p. 120, Example, line 2: no subscript on right side

p. 124, displayed calculation near bottom: f(y)^{-1} is better than f^{-1}(y).

p. 126, line 3 of proof of Prop. 4: "for all sufficiently small open normal..."

p. 129, last line of 2nd paragraph: missing right parenthesis

p. 138, Application: seems K'/K should be assumed separable (and likewise the irreducible equation at the end should be separable).

p. 175, line above 6.3: in middle replace N_{M/K} with N_{M/K}(J_M).

p. 189, line -14: = 1, not = 0.

p. 190, line -12: source of \psi_p is Q*_p, not Q^{mc}_p.

p. 195, middle displayed formula for \beta_1(a), in next term should be \epsilon_2 between \beta_2 and (infl b).

p. 197, case r=-2, line 6: \widehat{H}^{-2}, not H^{-2}. (Also, the verification that the displayed map is actually inverse to the Artin map seems to merit a tiny bit of explanation (using the identification of artin map with cup product against fundamental class in Serre's treatment of the local case).

p. 198, line -10: not really typo, but ought to give a reference for this duality theorem, such as Thm. 6.6 in Ch, XII of Cartan-Eilenberg.

p. 200, displayed expression (15): lower right should be A (= A^{ab}), not E^{ab}, and the right vertical arrow is really composition of \psi_K with V:G_K^{ab} ----> G_L^{ab} = A.

p. 201, section 12, end of para. 2: \psi_K rather than \psi_L.

p. 207, 2 or 3 lines below displayed formula for \psi_p(x): replace "determined by" with "trivial on".

p. 211, 3 lines above \Phi(s,\chi) displayed formula: "For primitive Dirichlet...."

p. 211, line -12: I think a_q being 0 or 1 cases should be swapped (see calculation of \zeta(f_{+/-}, +/-| |^s) on p. 317; the definition of \Phi(s,\chi) has also dropped some of the powers of \pi^{1/2} from those formulas, and this doesn't harm the statement of the functional equation but isn't quite "right" without them).

p. 212, line 14: -c_p, not +c_p.

p. 214 line 13 : should clarify that if \eta = 0 it means o(x).

p. 214, line 15: the definition of c is missing some symbols ([k:Q]), and the definitions of f and g should subtracting some stuff.

p. 215, line 6: displayed limit should be set equal to \ell.

p. 215, line -1: "absolute first degree relative to k".

p. 220, line 4: "we take as the local..."

p. 221, line -4: replace \frak{Q} with \frak{q}.

p. 222, line -8: \mu_0, not \mu, on left side.

p. 224, line -4: this is not really another way of arriving at a contradiction (distinct from citing linear independence of characters), since this argument basically is how one proves linear independence of characters over a field with characteristic not dividing the size of G.

p. 225, lines 1--3: this seems fishy, since the proposed procedure would involve inverting a gxg submatrix that depends on the choice of p. The Brauer argument below makes it all moot.

p. 230, line 7: "in k", not "in K".

p. 353, line 4: (b,\lambda)_v rather than (\lambda,b)_v.

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