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Algebraic Number Theory

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Algebraic Number Theory Algebraic Number Theory J.S. Milne Version 3.08 July 19, 2020 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. BibTeX information @misc{milneANT, author={Milne, James S.}, title={Algebraic Number Theory (v3.08)}, year={2020}, note={Available at www.jmilne.org/math/}, pages={166} } 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. v3.02 (April 30, 2009). Minor fixes; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. v3.04 (April 12, 2012). Minor fixes. v3.05 (March 21, 2013). Minor fixes. v3.06 (May 28, 2014). Minor fixes; 164 pages. v3.07 (March 18, 2017). Minor fixes; 165 pages. v3.08 (July 19, 2020). Minor fixes; 166 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at jmilne at umich dot edu. The photograph is of the Fork Hut, Huxley Valley, New Zealand. Copyright c 1996–2020 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Notation....................................... 4 Introduction..................................... 7 Exercises ...................................... 12 1 Preliminaries from Commutative Algebra 14 Basic definitions................................... 14 Ideals in products of rings.............................. 15 Noetherian rings................................... 15 Noetherian modules................................. 16 Local rings ..................................... 17 Rings of fractions.................................. 18 The Chinese remainder theorem .......................... 19 Review of tensor products.............................. 21 Exercise....................................... 24 2 Rings of Integers 25 First proof that the integral elements form a ring.................. 25 Dedekind’s proof that the integral elements form a ring.............. 26 Integral elements .................................. 28 Review of bases of A-modules........................... 31 Review of norms and traces............................. 31 Review of bilinear forms .............................. 32 Discriminants.................................... 33 Rings of integers are finitely generated....................... 35 Finding the ring of integers............................. 37 Algorithms for finding the ring of integers..................... 40 Exercises ...................................... 44 3 Dedekind Domains; Factorization 46 Discrete valuation rings............................... 46 Dedekind domains ................................. 48 Unique factorization of ideals............................ 49 The ideal class group................................ 52 Discrete valuations ................................. 55 Integral closures of Dedekind domains....................... 57 Modules over Dedekind domains (sketch)...................... 57 Factorization in extensions ............................. 59 The primes that ramify ............................... 60 1 Finding factorizations................................ 62 Examples of factorizations ............................. 63 Eisenstein extensions................................ 66 Exercises ...................................... 67 4 The Finiteness of the Class Number 68 Norms of ideals................................... 68 Statement of the main theorem and its consequences................ 70 Lattices ....................................... 73 Some calculus.................................... 77 Finiteness of the class number ........................... 79 Binary quadratic forms............................... 81 Exercises ...................................... 83 5 The Unit Theorem 85 Statement of the theorem.............................. 85 Proof that UK is finitely generated......................... 87 Computation of the rank .............................. 88 S-units........................................ 90 Example: CM fields................................. 90 Example: real quadratic fields ........................... 91 Example: cubic fields with negative discriminant ................. 92 Finding .K/ .................................... 93 Finding a system of fundamental units....................... 93 Regulators...................................... 94 Exercises ...................................... 94 6 Cyclotomic Extensions; Fermat’s Last Theorem. 95 The basic results................................... 95 Class numbers of cyclotomic fields.........................101 Units in cyclotomic fields..............................101 The first case of Fermat’s last theorem for regular primes . 102 Exercises ......................................104 7 Absolute Values; Local Fields 105 Absolute Values...................................105 Nonarchimedean absolute values..........................106 Equivalent absolute values .............................107 Properties of discrete valuations ..........................109 Complete list of absolute values for the rational numbers . 110 The primes of a number field............................111 The weak approximation theorem .........................113 Completions.....................................114 Completions in the nonarchimedean case......................116 Newton’s lemma ..................................119 Extensions of nonarchimedean absolute values...................123 Newton’s polygon..................................125 Locally compact fields ...............................126 Unramified extensions of a local field .......................127 Totally ramified extensions of K . 129 Ramification groups.................................130 Krasner’s lemma and applications .........................131 Exercises ......................................133 8 Global Fields 135 Extending absolute values..............................135 The product formula ................................137 Decomposition groups ...............................139 The Frobenius element ...............................141 Examples ......................................143 Computing Galois groups (the hard way)......................144 Computing Galois groups (the easy way)......................144 Applications of the Chebotarev density theorem..................149 Finiteness Theorems ................................151 Exercises ......................................152 A Solutions to the Exercises 153 B Two-hour examination 160 Bibliography 161 Index 163 Notation We use the standard (Bourbaki) notation: N 0;1;2;::: ; Z ring of integers; R field D f g D D of real numbers; C field of complex numbers; Fp Z=pZ field with p elements, p a D D D prime number. For integers m and n, m n means that m divides n, i.e., n mZ. Throughout the notes, j 2 p is a prime number, i.e., p 2;3;5;:::. D Given an equivalence relation, Œ denotes the equivalence class containing . The empty set is denoted by . The cardinality of a set S is denoted by S (so S is the number of ; j j j j elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i I , is a function i ai I A. X YX is2 a subset of Y (not7! necessarilyW ! proper); X def YX is defined to be Y , or equals Y by definition; D X YX is isomorphic to Y ; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism); ' , denotes an injective map; ! denotes a surjective map. monnnnn question nnnnn in mathoverflow.net It is standard to use Gothic (fraktur) letters for ideals: abcmnpqABCMNPQ abcmnpqABCMNPQ Prerequisites The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and multilinear algebra. An undergraduate number theory course will also be helpful. References In addition to the references listed at the end and in footnotes, I shall refer to the following of my course notes (available at www.jmilne.org/math/): FT Fields and Galois Theory, v4.61, 2020. GT Group Theory, v3.16, 2020. CFT Class Field Theory, v4.02, 2013. Acknowledgements I thank the following for providing corrections and comments for earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Baraksha; Omar Baratelli; Francesc Castella;` Kwangho Choiy; Dustin Clausen; Keith Conrad; Edgar Costa, Sean Eberhard, Paul Federbush; Marco Antonio Flores Mart´ınez; Nathan Kaplan, Nevin Guo; Georg Hein; Florian Herzig; Dieter Hink; Hau-wen Huang; Enis Kaya; Keenan Kidwell; Jianing Li; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lawrence Howe; Lars Kindler; Franz Lemmermeyer; Milan Malciˇ c;´ Siddharth Mathur; Bijan Mohebi; Yogesh More; Scott Mullane; Safak Ozden; Wai Yan Pong; Nicolas´ Sirolli; Sam Spiro; Thomas Stoll; Bhupendra Nath Tiwari; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u-
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