Beginner's text for Algebraic Number Theory
What's a good book for learning Algebraic Number Theory with minimum prerequisites? Assume that the reader has done a basic abstract algebra course.
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$\begingroup$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.
$\endgroup$ 1 $\begingroup$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).
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ 3 $\begingroup$I liked Algebraic Theory of Numbers by Pierre Samuel.
$\endgroup$ $\begingroup$Paulo Ribenboim, A Classical Introduction to Algebraic Numbers, Springer 2001 " The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics. "
$\endgroup$ $\begingroup$There are notes by William A. Stein, in particular if you like to use computer to experiment:
$\endgroup$ $\begingroup$I like Algebraic Number Theory, Second Edition by Richard A. Mollin. It is simple and best book to self study.
$\endgroup$ $\begingroup$I recommend Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory by Harold M. Edwards.
I quote:
$\endgroup$ $\begingroup$This introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem" follows its historical development, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization. The more elementary topics, such as Euler's proof of the impossibilty of $x^3 +y^3 = z^3$, are treated in an uncomplicated way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's theory to quadratic integers and relates this to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.
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.
$\endgroup$ 1 $\begingroup$I want to recommend the Neukirch book "Algebraic Number Theorey". It is a good book.
$\endgroup$ $\begingroup$Quadratic Number Theory：An Invitation to Algebraic Methods in the Higher Arithmetic by J. L. Lehman, I read this as supplementary materials of cox's x^2+ny^2 and Ireland's book
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$The book Introductory algebraic number theory by Saban Alaca, Kenneth S. Williams is a good choice to start studying algebraic number theory.
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