Ebook: Theory of Algebraic Integers

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.