# Modern definition of a field

1. Feb 27, 2014

### Stephen Tashi

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

2. Feb 27, 2014

### 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”.