A field is any set of elements which satisfies the Field Axioms for both addition and multiplication and is a commutative Division Algebra. An archaic name for a field is Rational Domain. A field with a finite number of members is known as a Finite Field or Galois Field.

Because the identity condition must be different for addition and multiplication, every field must have at
least two elements. Examples include the Complex Numbers (), Rational
Numbers (), and Real Numbers (), but *not* the
Integers (), which form a Ring. It has been proven by Hilbert and
Weierstraß that all generalizations of the field concept to triplets of elements are
equivalent to the field of Complex Numbers.

© 1996-9

1999-05-26