What's a good book for learning Algebraic Number Theory with minimum prerequisites? Assume that the reader has done a basic abstract algebra course.
2026-04-09 12:37:16.1775738236
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Beginner's text for Algebraic Number Theory
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I'm a big fan of Murty and Esmonde's "Problems in algebraic number theory", which develops the basic theory through a series of problems (with the answers in the back). Another nice source is Milne's notes on algebraic number theory, available on his website here.
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I would recommend Stewart and Tall's Algebraic Number Theory and Fermat's Last Theorem for an introduction with minimal prerequisites. For example you don't need to know any module theory at all and all that is needed is a basic abstract algebra course (assuming it covers some ring and field theory). But again the first chapter gives a quick review of the basic material from abstract algebra that is used later on.
I have used this book when I took an introductory course in Algebraic Number Theory and the experience was really good. The book has many examples and the pace is not too fast.
Most other books I have seen rely more heavily on module theory to make the exposition more general but since probably you don't have that background yet then I guess that this book may very well suit you perfectly.
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There are notes by William A. Stein, in particular if you like to use computer to experiment: https://wstein.org/books/ant/ant.pdf
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Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic by J. L. Lehman. I read this as supplementary material to Cox's Primes of the Form $x^2+ny^2$ and Ireland & Rosen's A Classical Introduction to Modern Number Theory.
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Jarvis's Algebraic Number Theory is a gentle, undergraduate introduction. It is certainly accessible with basic abstract algebra - groups up to Lagrange's theorem and some familiarity with rings and fields are enough to get started, along with linear algebra and a little elementary number theory.
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Paul Pollack, A Conversational Introduction to Algebraic Number Theory: Arithmetic Beyond $\bf Z$
From the American Math Society website:
Gauss famously referred to mathematics as the “queen of the sciences” and to number theory as the “queen of mathematics”. This book is an introduction to algebraic number theory, meaning the study of arithmetic in finite extensions of the rational number field $\bf Q$. Originating in the work of Gauss, the foundations of modern algebraic number theory are due to Dirichlet, Dedekind, Kronecker, Kummer, and others. This book lays out basic results, including the three “fundamental theorems”: unique factorization of ideals, finiteness of the class number, and Dirichlet's unit theorem. While these theorems are by now quite classical, both the text and the exercises allude frequently to more recent developments.
In addition to traversing the main highways, the book reveals some remarkable vistas by exploring scenic side roads. Several topics appear that are not present in the usual introductory texts. One example is the inclusion of an extensive discussion of the theory of elasticity, which provides a precise way of measuring the failure of unique factorization.
The book is based on the author's notes from a course delivered at the University of Georgia; pains have been taken to preserve the conversational style of the original lectures.
Readership: Undergraduate and graduate students interested in algebraic number theory.
I would recommend Ireland and Rosen's text Classical Introduction to Modern Number Theory. The prerequisites for the first 10 or so chapters are minimal (first semester undergraduate algebra course).