- ALGEBRAIC STRUCTURES 451
2. Algebraic structures
The concept of a group was the first of the many abstractions that make up the
world of modern algebra. We have seen how it arises through the study of the
permutations of the roots of an equation. In the work of Lagrange, Ruffini, Cauchy,
and Abel, only the number of different forms that a function of the roots could take
was studied. Then Galois focused attention on the structure of the permutations
themselves, and the result was the first abstract structure, a permutation group.
Another two decades passed before the idea of a group was made abstract by Arthur
Cayley (1821-1895) in 1849. Cayley defined a group as a set of symbols that could
be combined in a way that was associative (he used the word) but not necessarily
commutative, and such that the elements must repeat themselves if all are operated
on by the same element. (From Cayley's language it is not clear whether he intended
this last property as an axiom or believed that it followed from the other properties
of a group.) An important example given by Cayley was a group of matrices.^13
The complete set of axioms for an abstract group was stated by Walther von Dyck
(1856-1934), a student of Felix Klein, in 1883.2.1. Fields, rings, and algebras. The concept of a group arose in the study of
the procedures used to solve equations, but that study involved other concepts that
were also somewhat vague and in need of clarification. What exactly did Galois
mean when he said "if we agree to regard certain objects as known" and spoke
of adjoining roots to an equation? Rational expressions in variables representing
unspecified numbers had long been part of the discourse in the solution of algebraic
equations. Both Abel and Galois made frequent use of this concept. Over the course
of the nineteenth century this domain of rationality evolved into what Dedekind in
1858 called a Zahlkdrper (number body). Leopold Kronecker (1823-1891) preferred
the term Rationalitats-Bereich (domain of rationality). The abstract object that
grew out of this concept eventually came to be known in French as a corps, in
German as a Korper, and in English as a field.14, Dedekind considered only fields
built on top of the rational numbers. Finite fields were first introduced, along
with the word field itself, by Å. H. Moore (1862-1932) in a paper published in the
Bulletin of the New York Mathematical Society in 1893.
Other algebraic concepts arise as generalizations of number systems. In partic-
ular, the integers, the complex numbers of the form m + ni (the Gaussian integers),
and the integers modulo a fixed integer m led to the general concept that Hubert,
in his exhaustive 1897 report to the German Mathematical Union "Die Theorie
des allgemeinen Zahlkorpers" ("The theory of the general number field") , called a
Zahlring (Number ring). He gave as an equivalent term Integritatsbereich (integral
domain). Both, however, were names for sets of complex numbers. Nowadays, an
integral domain is defined abstractly as a commutative ring with identity in which
the product of nonzero elements is nonzero; that is, there do not exist zero divisors.
An element that is not a zero divisor is said to be regular. These structures were
consciously abstracted and developed into the concept of an abstract ring in the
paper "Uber die Teiler der Null und die Zerlegung von Ringen" ("On zero divisors
and the decomposition of rings") by Adolf Fraenkel (1891-1965), which appeared(^13) The word matrix was Cayley's invention; the word is Latin for womb and is used figuratively in
mining to denote an ore-bearing rock. Cayley's "wombs" bore numbers rather than ore or babies.
(^14) A corps, however, is not necessarily commutative. Strictly speaking, it corresponds to what is
called in English a division ring.