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Local Fields: 67 (Graduate Texts in Mathematics) Hardcover – 27 Jul 1995


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A ring A is called a discrete valuation ring if it is a principal ideal domain (Boarbaki, Alg., Chap. VII) that has a unique non-zero prime ideal m(A). Read the first page
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Amazon.com: 1 review
16 of 20 people found the following review helpful
Elegant, but with Many misprints 2 May 2008
By IB - Published on Amazon.com
Format: Hardcover
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.
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