How Did the Definition of a Field Evolve to Require Commutativity?

[Stephen Tashi]

In the book Galois Theory by Emil Artin (2nd Ed 1965 of a work copyrighted 1942), he says

A field is a set of elements in which a pair of operations called multiplication and addition is defined analogous to the operations of multiplication and addition in the real number system (which is itself an example of a field). In each field F there exist unique elements called 0 and 1 which, under the operations of addition and multiplication behave with respect to all other elements of F exactly as their correspondents in the real number system. In two respects, the analogy is not complete: 1) multiplication is not assumed to be commutative in every field, and 2) a filed may have only a finite number of elements.

By contrast the modern definition of a field is that it is a commutative ring in which each nonzero element has a multiplicative identity. What developments caused the change in the definition with respect to commutativity?

 

[mathman]

http://en.wikipedia.org/wiki/Division_ring

from above:

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring[1] in which every nonzero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements.

Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.