Number Theory: Algebraic Numbers and FunctionsAmerican Mathematical Soc., 2000 - 368 pagina's Algebraic number theory is one of the most refined creations in mathematics. It has been developed by some of the leading mathematicians of this and previous centuries. The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory. Following the example set for us by Kronecker, Weber, Hilbert and Artin, algebraic functions are handled here on an equal footing with algebraic numbers. This is done on the one hand to demonstrate the analogy between number fields and function fields, which is especially clear in the case where the ground field is a finite field. On the other hand, in this way one obtains an introduction to the theory of 'higher congruences' as an important element of 'arithmetic geometry'. Early chapters discuss topics in elementary number theory, such as Minkowski's geometry of numbers, public-key cryptography and a short proof of the Prime Number Theorem, following Newman and Zagier. Next, some of the tools of algebraic number theory are introduced, such as ideals, discriminants and valuations. These results are then applied to obtain results about function fields, including a proof of the Riemann-Roch Theorem and, as an application of cyclotomic fields, a proof of the first case of Fermat's Last Theorem. There are a detailed exposition of the theory of Hecke $L$-series, following Tate, and explicit applications to number theory, such as the Generalized Riemann Hypothesis. Chapter 9 brings together the earlier material through the study of quadratic number fields. Finally, Chapter 10 gives an introduction to class field theory. The book attempts as much as possible to give simple proofs. It can be used by a beginner in algebraic number theory who wishes to see some of the true power and depth of the subject. The book is suitable for two one-semester courses, with the first four chapters serving to develop the basic material. Chapters 6 through 9 could be used on their own as a second semester course. |
Inhoudsopgave
1 | |
The Geometry of Numbers | 35 |
Dedekinds Theory of Ideals | 65 |
Valuations | 103 |
Algebraic Functions of One Variable | 141 |
Normal Extensions | 171 |
LSeries | 203 |
Applications of Hecke LSeries | 259 |
Quadratic Number Fields | 275 |
What Next? | 315 |
Appendix A Divisibility Theory | 325 |
Appendix B Trace Norm Different and Discriminant | 341 |
Harmonic Analysis on Locally Compact | 345 |
359 | |
367 | |
Veelvoorkomende woorden en zinsdelen
algebraic number field arbitrary associated b₁ basis character class group class number coefficients compact congruence consider continued fraction converges decomposition Dedekind defined denotes discriminant equal exists exponent field F field of constants finite extension following theorem function field Furthermore Galois group Grössencharakter Haar measure Hecke L-series homomorphism idele class group Im/Sm inertia degree irreducible isomorphism Lemma mapping minimal polynomial modulo multiple natural number nonarchimedean nonzero notation number theory obtain p₁ prime divisor prime element prime ideal prime number principal ideal domain proof of Theorem Proposition prove quadratic number field quasi-character ramification group ramification index ramified ray class ray class group real numbers relatively prime residue class field respect ring roots of unity Section sequence subgroup subset uniquely determined unramified zero zeta function απ
Populaire passages
Pagina 1 - God created the natural numbers; all else is the work of man.
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