The concepts of field, commutative ring, ideal, and unique factorization are among the fundamental notions of commutative algebra. How did they arise? In large measure, from three central problems in number theory: Fermat's Last Theorem, reciprocity laws, and binary quadratic forms. In this paper we will describe how this happened.
To put the issues in a broader context, these three number-theoretic problems were instrumental in the emergence of algebraic number theory—one of the two main sources of the modern discipline of commutative algebra. The other source was algebraic geometry. It was in the setting of these two subjects that many of the main concepts and results of commutative algebra evolved in the nineteenth century. Thus commutative algebra can be said to have been well developed before it was created.
Commutative algebra is nowadays understood to be the study of commutative noetherian rings. The first book on the subject, dating from the 1950s, was Zariski and Samuel's Commutative Algebra. In the first decades of the twentieth century, what we understand as commutative algebra was known as Ideal Theory, a name which reflects the sources of the subject.
To set the scene for our story, a word about mathematics, and especially algebra, in the nineteenth century. The period witnessed fundamental transformations in mathematics—in its concepts, its methods, and in mathematicians' attitude toward their subject.