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Number theory. Transl. from the 3rd German edition, edited and with a preface by Horst Günter Zimmer. Reprint of the 1980 edition. (English) Zbl 0991.11001

Classics in Mathematics. Berlin: Springer. xvii, 638 p. EUR 34.95/net; sFr. 58.00; £24.50; $ 39.95 (2002).
This is the reprint of the 1979 edition of ‘Number Theory’, H. G. Zimmer’s translation of Hasse’s well known Zahlentheorie (see Zbl 0423.12001). Hasse’s aim was the presentation of his view of number theory, in which number fields and function fields (of one variable with a finite field of constants) are studied simultaneously whenever possible. Hasse starts by studying the standard topics in elementary number theory (divisibility, congruences, residue class rings, quadratic reciprocity) both for the field of rational numbers and for the rational function field; the fact that the proofs in the function field case are usually given as an ‘appendix’ with instructions on how to modify the proofs in the number field case means that novices have to be disciplined enough to carry out the proofs for themselves; for readers who need more guidance, M. Rosen’s new book [Number theory in function fields. Graduate Texts in Mathematics 210. Springer, New York (2002; Zbl 1043.11079)] might be a better place to start learning the function field side of (algebraic) number theory.
Hasse then presents the theory of valuations, arithmetic in valued fields and their completions, the theory of divisors, differents and discriminants, and ends with a few chapters on important examples: quadratic and cyclotomic extensions (of the field of rational numbers), and the computation of units and the class number.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Rxx Algebraic number theory: global fields
11Sxx Algebraic number theory: local fields
12J10 Valued fields
11-02 Research exposition (monographs, survey articles) pertaining to number theory
12J25 Non-Archimedean valued fields
12J20 General valuation theory for fields
11R58 Arithmetic theory of algebraic function fields
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