Modern definition of a field

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

Quote:

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?