 
Summary: Dedekind's 1871 version of
the theory of ideals
Translated by Jeremy Avigad
March 19, 2004
Translator's introduction
By the middle of the nineteenth century, it had become clear to mathe
maticians that the study of finite field extensions of the rational numbers is
indispensable to number theory, even if one's ultimate goal is to understand
properties of diophantine expressions and equations in the ordinary integers.
It can happen, however, that the "integers" in such extensions fail to satisfy
unique factorization, a property that is central to reasoning about the or
dinary integers. In 1844, Ernst Kummer observed that unique factorization
fails for the cyclotomic integers with exponent 23, i.e. the ring Z[] of inte
gers of the field Q(), where is a primitive twentythird root of unity. In
1847, he published his theory of "ideal divisors" for cyclotomic integers with
prime exponent. This was to remedy the situation by introducing, for each
such ring of integers, an enlarged domain of divisors, and showing that each
integer factors uniquely as a product of these. He did not actually construct
these integers, but, rather, showed how one could characterize their behav
ior qua divisibility in terms of ordinary operations on the associated ring of
