# Algebraic Number Theory

• Frazer Jarvis
Textbook

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

1. Front Matter
Pages i-xiii
2. Frazer Jarvis
Pages 1-16
3. Frazer Jarvis
Pages 17-37
4. Frazer Jarvis
Pages 39-63
5. Frazer Jarvis
Pages 65-86
6. Frazer Jarvis
Pages 87-112
7. Frazer Jarvis
Pages 113-147
8. Frazer Jarvis
Pages 149-168
9. Frazer Jarvis
Pages 169-189
10. Frazer Jarvis
Pages 191-206
11. Frazer Jarvis
Pages 207-230
12. Frazer Jarvis
Pages 231-255
13. Back Matter
Pages 257-292

### Introduction

The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic.

Algebraic Number Theory takes 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.

### Keywords

Algebraic Number Theory Class Groups Number Field Sieve Quadratic Fields Unique Factorisation

#### Authors and affiliations

• Frazer Jarvis
• 1
1. 1.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUnited Kingdom

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-07545-7
• Copyright Information Springer International Publishing Switzerland 2014
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-07544-0
• Online ISBN 978-3-319-07545-7
• Series Print ISSN 1615-2085
• Series Online ISSN 2197-4144
• Buy this book on publisher's site
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