Product Description

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.

Customer Reviews (2)

Good Book!
If you are the type of person who likes essayistic portrayals of mathematical theories and hence finds yourself tending toward those books that waste your time explaining every little detail, then yes you will not enjoy this book.Ribenbiom's book, like that of Shafarevich and Neukirch, is in a league of its own and is un-paralleled by any of the more recent texts that have become available over the last five to ten years in the field of algebraic number theory.This book is, just as it should be, a treatise on algebraic number theory which is meant for researchers and dedicated, as well as, determined students.This is not intended for those interested in applications to cryptography or computer science or etc.
Now, let me assert that I am definitely not an expert in the field, just an admirer, but with respect to the flow of the book I can say that in terms of content the direction is fairly standard, whereas, relative to reading this book, it is very easy to get bogged down and spend several days on one single page and, yes, this can be incredibly frustrating but this should not relegate the book to a position of being called "a bad book."If you have time, motivation, more time and enjoy number theory, then this book is a must and, in particular, if you are a blossoming mathematician and working towards an advanced degree in algebraic number theory, then this book must be read since it contains more than enough to shape you into a skilled and proficient number theorist.
Given such a review, you may be scratching your head and wondering why I have rated this book four instead of five stars.The reason is simple, I opted for four stars since I have not read this book in its entirety (just over three-quarters) and because of this if I had opted to give a rating of five stars, then I could be accused, erroneously, of some type of favoritism.Enjoy!

Bad book!
This book is bad! To make it clear, the author seems to be an expert in the field, so the results are stated correctly, but not in a correct way. Definition followed by theorems, then by proofs, then by exercises. No explaination, no motivation, no flow, no organism, makes a beautiful subject so painful to proceed.

On the positive side, the book has good scope and depth for an introductory text, along with a descent set of exercises. That makes some value as a reference book. But if you want to learn algebraic number theory, stay away from this book. ... Read more

Product Description
In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory. The finite fields are classified by size; there is exactly one finite field up to isomorphism of size pk for each prime p and positive integer k. Each finite field of size q is the splitting field of the polynomial xq - x, and thus the fixed field of the Frobenius endomorphism which takes x to xq. Similarly, the multiplicative group of the field is a cyclic group. Wedderburn's little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field. Finite fields have applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type. Finite fields are an active area of research, including recent results on the Kakeya conjecture and open problems on the size of the smallest primitive root. ... Read more

Product Description
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localisation techniques lead naturally from abelian varieties defined over number fields to ones defined over finite fields and various local fields. ... Read more