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Algebraic Number Theory [Jarvis] - mklabgr - 08-17-2026 Algebraic Number Theory Author: Frazer Jarvis Publication date: 2014 Publisher: Springer Series: Springer Undergraduate Mathematics Series Frazer Jarvis's Algebraic Number Theory is designed as an accessible undergraduate introduction to a subject that is often presented at graduate level. Jarvis begins with the familiar fact that positive integers have unique prime factorizations and then asks what happens when arithmetic is extended from $\mathbb Z$ and $\mathbb Q$ to larger number fields. The crucial discovery is that ordinary unique factorization can fail in rings of algebraic integers. This failure motivates one of the central ideas of the subject: ideals. Jarvis develops number fields, discriminants, integral bases, ideals and prime ideals, showing how unique factorization can effectively be recovered at the level of ideals. Quadratic fields provide many of the concrete examples, allowing the reader to perform explicit calculations while learning the general theory.  The book then expands into more sophisticated aspects of algebraic number theory. Jarvis studies imaginary quadratic fields, lattices and geometric methods, fields of small degree, and cyclotomic fields, including their relationship with the Fermat equation. The later chapters introduce analytic techniques and the analytic class number formula before culminating in the number field sieve, one of the most important modern algorithms for factoring large integers. This final topic is particularly distinctive: Springer notes that Jarvis introduces the number field sieve at a level suitable for undergraduates, connecting classical algebraic number theory with modern computational number theory.  An important strength of the book is its pedagogical approach. It requires relatively modest algebraic prerequisites and introduces ideals through their historical motivation—the breakdown of unique factorization—rather than presenting them simply as abstract algebraic objects. Examples and exercises are used extensively, with hints and solutions included. This makes the book suitable not only for an undergraduate course but also for self-study by someone who already knows basic abstract algebra and elementary number theory.  Key takeaways Unique factorization drives the story. The book starts with prime factorization in $\mathbb Z$ and uses its failure in number fields to motivate much of algebraic number theory. Ideals repair the problem. Even when algebraic integers do not factor uniquely into elements, ideals admit the appropriate form of unique prime factorization. Concrete examples lead to abstraction. Quadratic fields are used extensively so that concepts such as discriminants, integral bases and ideals can actually be calculated. Classical theory meets computation. The progression from number fields and cyclotomic fields to analytic methods and the number field sieve gives the book an unusually broad undergraduate scope. Overall: Jarvis's book is a very good choice for a first serious course in algebraic number theory. Compared with advanced classics such as Lang or Neukirch, its purpose is not maximum generality but accessibility: it builds the theory from familiar arithmetic, provides plenty of concrete examples, and gradually moves toward genuinely advanced ideas. For a mathematics graduate or strong undergraduate wanting a bridge from elementary number theory and abstract algebra into modern algebraic number theory, it is particularly well positioned. Goodreads – Algebraic Number Theory by Frazer Jarvis Springer – Algebraic Number Theory by Frazer Jarvis