Authors:
- Provides a self-contained and easy-to-read introduction to algebraic number theory, with minimal algebraic prerequisites
- Introduces the theory of ideals in a historical context, through the study of the failure of unique factorisation in number fields
- Introduces the number field sieve at a level suitable for undergraduates
- Includes supplementary material: sn.pub/extras
Part of the book series: Springer Undergraduate Mathematics Series (SUMS)
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Table of contents (11 chapters)
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Front Matter
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Back Matter
About this book
This undergraduate textbook provides an approachable and thorough introduction to the topic of algebraic number theory, taking the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform.
The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the first time that the number field sieve has been considered in a textbook at this level.
Reviews
“Undergraduate mathematics students need both to develop facility with numerical and symbolic calculation and comfort with abstraction. Algebraic number theory offers an ideal context for encountering the synthesis of these goals. One could compile a shelf of graduate-level expositions of algebraic number theory, and another shelf of undergraduate general number theory texts that culminate with a first exposure to it. … Summing Up: Highly recommended. Upper-division undergraduates.” (D. V. Feldman, Choice, Vol. 52 (8), April, 2015)
“In this book, the author leads the readers from the theorem of unique factorization in elementary number theory to central results in algebraic number theory. … This book is designed for being used in undergraduate courses in algebraic number theory; the clarity of the exposition and the wealth of examples and exercises (with hints and solutions) also make it suitable for self-study and reading courses.” (Franz Lemmermeyer, zbMATH, Vol. 1303, 2015)Authors and Affiliations
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School of Mathematics and Statistics, University of Sheffield, Sheffield, United Kingdom
Frazer Jarvis
About the author
Bibliographic Information
Book Title: Algebraic Number Theory
Authors: Frazer Jarvis
Series Title: Springer Undergraduate Mathematics Series
DOI: https://doi.org/10.1007/978-3-319-07545-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2014
Softcover ISBN: 978-3-319-07544-0Published: 04 July 2014
eBook ISBN: 978-3-319-07545-7Published: 23 June 2014
Series ISSN: 1615-2085
Series E-ISSN: 2197-4144
Edition Number: 1
Number of Pages: XIII, 292
Number of Illustrations: 3 b/w illustrations