Algebraic Number Theory - James Milne

Sep 28, 2008 - note={Available at www.jmilne.org/math/}, .... Integral closures of Dedekind domains . ... Modules over Dedekind domains (sketch).
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Algebraic Number Theory

J.S. Milne

Version 3.01 September 28, 2008 A more recent version of these notes is available at www.jmilne.org/math/

An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. The original version was distributed during the teaching of a second-year graduate course.

BibTeX information @misc{milneANT, author={Milne, James S.}, title={Algebraic Number Theory (v3.01)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={155+viii} }

v2.01 (August 14, 1996). First version on the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page.

The photograph is of the Fork Hut, Huxley Valley, New Zealand.

c Copyright 1996, 1998, 2008, J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder.

Contents Contents Notations. . . . . . Prerequisites . . . . References . . . . . Acknowledgements Introduction . . . . Exercises . . . . . 1

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2 5 5 5 5 1 6

Preliminaries from Commutative Algebra Basic definitions . . . . . . . . . . . . . . . Ideals in products of rings . . . . . . . . . . Noetherian rings . . . . . . . . . . . . . . . Noetherian modules . . . . . . . . . . . . . Local rings . . . . . . . . . . . . . . . . . Rings of fractions . . . . . . . . . . . . . . The Chinese remainder theorem . . . . . . Review of tensor products . . . . . . . . . . Exercise . . . . . . . . . . . . . . . . . . .

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7 7 8 8 9 10 11 12 14 17

Rings of Integers First proof that the integral elements form a ring . . . . Dedekind’s proof that the integral elements form a ring Integral elements . . . . . . . . . . . . . . . . . . . . Review of bases of A-modules . . . . . . . . . . . . . Review of norms and traces . . . . . . . . . . . . . . . Review of bilinear forms . . . . . . . . . . . . . . . . Discriminants . . . . . . . . . . . . . . . . . . . . . . Rings of integers are finitely generated . .