Algebraic Number Theory

Another concept of abstract algebra that emerged from the old algebra of equations was that of

ring

, which arose from attempts to find

integer

solutions of equations. The first steps towards the ring concept were taken by Euler (1770b), who discovered equations whose integer solutions are most easily found with the help of irrational or imaginary numbers. Gauss realized that these auxiliary numbers work because they

behave like

integers. In particular, they admit a concept of “prime” for which unique prime factorization holds. In the 1840s and 1850s the idea of “algebraic integers” was pushed further by various mathematicians, and it reached maturity when Dedekind (1871) defined the concept of

algebraic integer

in a

number field of finite degree

. By this time, considerable experience with number fields had been acquired, and Kummer had noticed that such fields do

not

always admit unique prime factorization. Kummer found a way around this difficulty by introducing new objects that he called

ideal numbers

(in analogy with “ideal” objects in geometry, such as points at infinity). Dedekind replaced Kummer’s undefined “ideal numbers” by concrete sets of numbers that he called

ideals

. He was then able to restore unique prime factorization by proving that it holds for ideals. Ring theory as we know it today is largely the result of building a general setting for Dedekind’s theory of ideals. It owes its existence to Emmy Noether, who used to say that “it’s already in Dedekind.”