"Algebraic Number Theory 001.ps.gz" - читать интересную книгу автораALGEBRAIC NUMBER THEORY J.S. MILNE Abstract. These are the notes for a course taught at the University of Michigan in F92 as Math 676. They are available at www.math.lsa.umich.edu/,jmilne/. Please send comments and corrections to me at [email protected]. v2.01 (August 14, 1996.) First version on the web. v2.10 (August 31, 1998.) Fixed many minor errors; added exercises and index. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The ring of integers 1; Factorization 2; Units 4; Applications 5; A brief history of numbers 6; References. 7. 1. Preliminaries from Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Basic definitions 10; Noetherian rings 10; Local rings 12; Rings of fractions 12; The Chinese remainder theorem 14; Review of tensor products 15; Extension of scalars 17; Tensor products of algebras 17; Tensor products of fields 17. 2. Rings of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Symmetric polynomials 19; Integral elements 20; Review of bases of Amodules 25; Review of norms and traces 25; Review of bilinear forms 26; Discriminants 26; Rings of integers are finitely generated 28; Finding the ring of integers 30; Algorithms for finding the ring of integers 33. Discrete valuation rings 37; Dedekind domains 38; Unique factorization 39; The ideal class group 43; Discrete valuations 46; Integral closures of Dedekind domains 47; Modules over Dedekind domains (sketch). 48; Factorization in extensions 49; The primes that ramify 50; Finding factorizations 53; Examples of factorizations 54; Eisenstein extensions 56. 4. The Finiteness of the Class Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Norms of ideals 58; Statement of the main theorem and its consequences 59; Lattices 62; Some calculus 67; Finiteness of the class number 69; Binary quadratic forms 71; 5. The Unit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Statement of the theorem 73; Proof that UK is finitely generated 74; Computation of the rank 75; S-units 77; Finding fundamental units in real cfl1996, 1998, J.S. Milne. You may make one copy of these notes for your own personal use. i 0 J.S. MILNE quadratic fields 77; Units in cubic fields with negative discriminant 78; Finding _(K) 80; Finding a system of fundamental units 80; Regulators 80; |
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