The title of this book may be read in two ways. One is 'algebraic number-theory', that is, the theory of numbers viewed algebraically; the other, 'algebraic-number theory', the study of algebraic numbers. Both readings are compatible with our aims, and both are perhaps misleading. Misleading, because a proper coverage of either topic would require more space than is available, and demand more of the reader than we wish to; compatible, because our aim is to illustrate how some of the basic notions of the theory of algebraic numbers may be applied to problems in number theory. Algebra is an easy subject to compartmentalize, with topics such as 'groups', 'rings' or 'modules' being taught in comparative isolation. Many students view it this way. While it would be easy to exaggerate this tendency, it is not an especially desirable one. The leading mathematicians of the nineteenth and early twentieth centuries developed and used most of the basic results and techniques of linear algebra for perhaps a hundred years, without ever defining an abstract vector space: nor is there anything to suggest that they suf- fered thereby. This historical fact may indicate that abstrac- tion is not always as necessary as one commonly imagines; on the other hand the axiomatization of mathematics has led to enormous organizational and conceptual gains.
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The origins of algebraic number theory.- I Algebraic Methods.- 1 Algebraic background.- 1.1 Rings and fields.- 1.2 Factorization of polynomials.- 1.3 Field extensions.- 1.4 Symmetric polynomials.- 1.5 Modules.- 1.6 Free abelian groups.- 2 Algebraic numbers.- 2.1 Algebraic numbers.- 2.2 Conjugates and discriminants.- 2.3 Algebraic integers.- 2.4 Integral bases.- 2.5 Norms and traces.- 3 Quadratic and cyclotomic fields.- 3.1 Quadratic fields.- 3.2 Cyclotomic fields.- 4 Factorization into irreducibles.- 4.1 Historical background.- 4.2 Trivial factorizations.- 4.3 Factorization into irreducibles.- 4.4 Examples of non-unique factorization into irreducibles.- 4.5 Prime factorization.- 4.6 Euclidean domains.- 4.7 Euclidean quadratic fields.- 4.8 Consequences of unique factorization.- 4.9 The Ramanujan-Nagell theorem.- 5 Ideals.- 5.1 Historical background.- 5.2 Prime factorization of ideals.- 5.3 The norm of an ideal.- II Geometric Methods.- 6 Lattices.- 6.1 Lattices.- 6.2 The quotient torus.- 7 Minkowski’s theorem.- 7.1 Minkowski’s theorem.- 7.2 The two-squares theorem.- 7.3 The four-squares theorem.- 8 Geometric representation of algebraic numbers.- 8.1 The space Lst.- 9 Class-group and class-number.- 9.1 The class-group.- 9.2 An existence theorem.- 9.3 Finiteness of the class-group.- 9.4 How to make an ideal principal.- 9.5 Unique factorization of elements in an extension ring.- III Number-Theoretic Applications.- 10 Computational methods.- 10.1 Factorization of a rational prime.- 10.2 Minkowski’s constants.- 10.3 Some class-number calculations.- 10.4 Tables.- 11 Fermat’s Last Theorem.- 11.1 Some history.- 11.2 Elementary considerations.- 11.3 Kummer’s lemma.- 11.4 Kummer’s Theorem.- 11.5 Regular primes.- 12 Dirichlet’s Units Theorem.- 12.1 Introduction.- 12.2 Logarithmic space.- 12.3 Embedding the unit group in logarithmic space.- 12.4 Dirichlet’s theorem.- Appendix 1 Quadratic Residues.- A.3 Quadratic Residues.- Appendix 2 Valuations.- References.
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