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Local fields and their extensions: a constructive approach. Translated from an original Russian manuscript by Ivan B. Fesenko; translation edited by Simeon Ivanov. (English) Zbl 0781.11042
Translations of Mathematical Monographs. 121. Providence, RI: American Mathematical Society (AMS). xv, 283 p. (1993).
This approach to the theory of local fields contains the main topics about complete discrete valuation fields with perfect residue class fields. Class field theory is presented for quasi-finite residue fields \(K\) (i.e. \(K\) is perfect to and to each \(n\in\mathbb N\) there is one and only one extension of \(K\) of degree \(n\)). The proofs use the direct methods of Hazewinkel and Neukirch avoiding the use of cohomology. Herbrand’s theory of higher ramification is based on the inverse Herbrand-function, called by the authors without further explanation “Hasse-Herbrand function”. This approach is helpful in the presentation of Fontaine’s and Wintenberger’s theory of norm fields. The last four chapters are devoted to the group of units, to explicit formulas for the Hilbert symbol, following Shafarevich and Vostokov, to the Hilbert pairing for formal groups and to Milnor \(K\) groups of local fields. There are two short appendices about the absolute Galois group of a local field and about multidimensional local fields.
The book is written by two mathematicians successfully working about local fields both one- and multidimensional. This determines its style and focal points. Lubin-Tate groups and their application to class field theory are “summarized” on four pages. There is an extensive bibliography which is very useful for every one interested in local fields.
The book is well written. The first four chapters present an introduction to the subject, while the following five chapters are devoted to more recent developments. A big amount of exercises contribute to the attraction of this highly original book.
Reviewer: H.Koch (Berlin)

11Sxx Algebraic number theory: local and \(p\)-adic fields
11S31 Class field theory; \(p\)-adic formal groups
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11S15 Ramification and extension theory
11S70 \(K\)-theory of local fields
11S99 Algebraic number theory: local and \(p\)-adic fields
11S20 Galois theory