Algebraic number theory, General, Integral Representations, Mathematics, Mathematics / Algebra / General, MATHEMATICS / Number Theory, Mathematics : Number Theory, Number theory, Science, Science / General, Science/Mathematics
The invention of ideals by Dedekind in the 1870s was well ahead of its time, and proved to be the genesis of what today we would call algebraic number theory. His memoir "Sur la Theorie des Nombres Entiers Algebriques" first appeared in installments in the Bulletin des sciences mathematiques in 1877. This book is a translation of that work by John Stillwell, who adds a detailed introduction giving historical background and who outlines the mathematical obstructions that Dedekind was striving to overcome. Dedekind's memoir offers a candid account of the development of an elegant theory and provides blow by blow comments regarding the many difficulties encountered en route. This book is a must for all number theorists.
Summary: An antidote against too much modernistic algebra
Rating: 5
Algebraic number theory is about employing unique factorisation in rings larger than the integers. The classical cases are the quadratic integers and the cyclotomic integers. They came with elaborate theories to deal with the fact that unique factorisation does not always hold. Dedekind generalises and cleans up these theories by developing a general theory of algebraic integers. Kummer's theory of ideal prime factors, which saved unique factorisation in some cases in the cyclotomic integers, is replaced by a beautifully conceptual and streamlined theory of ideals. The power of abstraction has perhaps never been more impressive. Many insights that today are scattered in abstract algebra and linear algebra can be seen here in their original glory, introduced not as soulless axiomatic structures but for their original noble purpose of understanding numbers.
Half the book consists of Stillwell's introduction, which is a brilliant sketch of the history of number theory from Diophantus to Dedekind, of course focusing especially on the prehistory of algebraic number theory.
Summary: Emmy Noether called it a must-read
Rating: 5
This is Dedekind's famous creation of the theory of (algebraic number) rings and modules, which he presented as an appendix to his edition of Dirichlet's LECTURES ON NUMBER THEORY. In fact it went through several editions, and the translation here is from another article he wrote to make the ideas more accessible. Anyway Noether had her students read every versiom of it. Her watchword was "It is all already in Dedekind", meaning largely this work. And she was right, in a very deep sense the whole modern approach to abstract algebra is in Dedekind, though it took her phenomenal genius to *find* it there.
Dedekind (most of the time) explicitly limits himself to modules of algebraic numbers, but Noether correctly saw that Dedekind already knew (many of) his theorems held for the whole abstract range she would explicate and develop. Benefitting from her, we can even see this generality peeking through in some of his remarks.
Anyone knowing the basic modern ideas of rings and modules can read this with pleasure, both as the origins of abstract algebra with many fine insights to offer, and as a connection to the concrete motives. Of course Dedekind wrote for people who did not know such things. But he assumed they would think very, very hard. He also assumed some arithmetic ideas not widely taught today, but nicely explained in Stillwell's preface. Dedekind is a wonderful writer, well served here by a clear translation. You are apt to fall in love with this book, and want to accompany it with Dirichlet's own LECTURES ON NUMBER THEORY, written up by Dedekind, and also translated to english by Stillwell.
