Local Fields (Graduate Texts in Mathematics) [Hardcover]

Jean-Pierre Serre (Author), Marvin J. Greenberg (Translator)

Book Description

ISBN-10: 0387904247 | ISBN-13: 978-0387904245 | Publication Date: January 19, 1980

This is a translation of Corps Locaux, supplemented with an updated bibliography and some new exercises. This book is accessible to graduate students, and can be used as a reference source by research mathematicians in algebra and number theory.

Editorial Reviews

Language Notes

Text: English (translation)
Original Language: French

Product Details

• Hardcover: 268 pages
• Publisher: Springer (January 19, 1980)
• Language: English
• ISBN-10: 0387904247
• ISBN-13: 978-0387904245
• Product Dimensions: 9.2 x 6.2 x 0.8 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,025,686 in Books (See Top 100 in Books)

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12 of 16 people found the following review helpful:

4.0 out of 5 stars Elegant, but with Many misprints, May 1, 2008

By 

IB "IB" (Houston, TX United States) - See all my reviews

This review is from: Local Fields (Graduate Texts in Mathematics) (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.