Local Fields (Graduate Texts in Mathematics, 67)

The mathematical content and exposition are at a high level typical of Serre.I have not finished reading the entire book, but here are some misprints I have found that may serve as a useful warning. NB: most of these errors are Not in the 3rd French edition...Chapter 1:section 4, pg. 14, 2nd centered display: the ramification indices should be e_{\beta} not e_{p} in the product.section 5, pg. 15, first formula needs to be N: I_{B}-> I_{A}, Not the other way around.section 6, pg. 17, last sentence of first paragraph, replace the inclusion symbol $\in$ with the word "in". Clearly, f is an element of A[X] and not an element of k[X]. In the French ed. Serre correctly used "dans" and did not us the symbol $\in$.section 7, pg. 22, in proof of Prop. 21, 2nd paragraph, 3rd sentence, replace "contain" with "contains".4th sentence: should be, "... we must have \bar{L}_{S} = \bar{K}_{T}" not \bar{L}. [separable consequence is later, namely in the Corollary(!)]Chapter 2:sec 1, pg. 28: third sentence should be "one sees that E is the union of (A:xA) cosets of modules xE,...". As is in the book, the sentence does not make grammatical sense.sec 2, pg. 29: the def. of w must carry a v' not just v, that is: w = (1/m) v' is a discrete valuation of L.sec 3, theorem 1, (i): change K to \hat{K}; so the completion of L_i has degree n_i over the completion of K.sec 3, exercise 1: the suggested reference should say Section 3 of Bourbaki Algebra, not 7. (going by Hermann Paris 1958 as usual)Chapter 4:sec 1, pg. 63, prop 3, need K' (not K) in def. of e', that is: e' = e_{L/K'}.in the proof of prop 3, the s and t for "st, t in H" need to be italicized.sec 2, prop 6, first line of proof: gothic beta should be gothic p, that is to each x in p^{i}_{L}sec 3, lemma 3, last line of proof: upper case Phi is nowhere defined, need lower case phi, that is: phi'(u)....so theta and phi must coincide.sec 3, statement of lemma 5, again phi, not Phi.Some tips for the beginner:- Know how localization behaves as a functor via, say Atiyah-Macdonald.- For a clean and clear proof that separable <=> nondegenerate Tr(,) see Roman's "Field Theory" (Bourbaki uses etale algebras to get this result, a bit more than needed).- P. Samuel's "Algebraic theory of numbers" (Dover publ. now!) has a very elegant exposition of the proof of quadratic reciprocity that is alluded to at the end of section 8.

This rating is not based on the mathematical content in this book – I would recommend Serre's influential and classic account to any willing reader. My low rating is due solely to the quality of the text. I was slightly shocked to receive a textbook more than twice the size (see photo) of most books (indeed, all that I have seen) published by Springer. I am at a loss as to why they printed it in such a way, and am quite frustrated that I will need to carry around this oversized version of a real masterpiece in the mathematics literature.

日本国内印刷のpaperbackの場合は、７０頁にPDFのアイコンのようなマークが印刷されてしまって、本文と重なって読めなくなっています。オンデマンド出版のデータの不具合かと思われます。返品して交換しましたが、交換した商品にも、もちろん、同じミスプリントがありました。

2018年の4月18日にAmazonでこの本(ペーパーバック)を購入しましたが、他の方のレビューにあるような不具合はどこにも見当たりませんでした。非常に良い本です。