# Number Theory: Algebraic Numbers and Functions (Graduate Studies in Mathematics)

## Helmut Koch

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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.

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A significant amount of the material goes beyond what one would expect from an introductory text. -- European Mathematical Society Newsletter

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## 1.Number Theory: Algebraic Numbers and Functions (Graduate Studies in Mathematics)

Editore: American Mathematical Society (2000)
ISBN 10: 0821820540 ISBN 13: 9780821820544
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Descrizione libro American Mathematical Society, 2000. Hardcover. Condizione libro: New. Codice libro della libreria DADAX0821820540

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## 2.Number Theory: Algebraic Numbers and Functions (Graduate Studies in Mathematics)

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Descrizione libro American Mathematical Society, 2000. Hardcover. Condizione libro: New. book. Codice libro della libreria 0821820540

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## 3.Number Theory: Algebraic Numbers and Functions (Graduate Studies in Mathematics)

Editore: American Mathematical Society (2000)
ISBN 10: 0821820540 ISBN 13: 9780821820544
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Descrizione libro American Mathematical Society, 2000. Hardcover. Condizione libro: New. Brand new. We distribute directly for the publisher. 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. Codice libro della libreria 1005120063

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## 4.Number Theory: Algebraic Numbers and Functions (Graduate Studies in Mathematics)

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Descrizione libro American Mathematical Society, 2000. Hardcover. Condizione libro: New. Codice libro della libreria P110821820540

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## 5.Number Theory

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Descrizione libro American Mathematical Society, 2000. HRD. Condizione libro: New. New Book. Shipped from UK in 4 to 14 days. Established seller since 2000. Codice libro della libreria CE-9780821820544

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## 6.Number Theory: Algebraic Numbers and Functions (Hardback)

Editore: American Mathematical Society, United States (2000)
ISBN 10: 0821820540 ISBN 13: 9780821820544
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Descrizione libro American Mathematical Society, United States, 2000. Hardback. Condizione libro: New. Language: English . Brand New Book. 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. Codice libro della libreria AAN9780821820544

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## 7.Number Theory: Algebraic Numbers and Functions (Hardback)

Editore: American Mathematical Society, United States (2000)
ISBN 10: 0821820540 ISBN 13: 9780821820544
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Descrizione libro American Mathematical Society, United States, 2000. Hardback. Condizione libro: New. Language: English . Brand New Book. 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. Codice libro della libreria AAN9780821820544

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## 8.Number Theory: Algebraic Numbers and Functions

Editore: American Mathematical Society
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Descrizione libro American Mathematical Society. Hardback. Condizione libro: new. BRAND NEW, Number Theory: Algebraic Numbers and Functions, Helmut Koch, 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. Codice libro della libreria B9780821820544

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## 9.Number Theory: Algebraic Numbers and Functions

Editore: Amer Mathematical Society (2000)
ISBN 10: 0821820540 ISBN 13: 9780821820544
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Descrizione libro Amer Mathematical Society, 2000. Hardcover. Condizione libro: Brand New. 368 pages. 10.25x7.25x1.00 inches. In Stock. Codice libro della libreria __0821820540

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## 10.Graduate Studies in Mathematics: Number Theory : Algebraic Numbers and Functions 24

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Descrizione libro Condizione libro: Brand New. Book Condition: Brand New. Codice libro della libreria 97808218205441.0

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