A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division
algebra. An archaic name for a field is rational domain. The French term for
a field is corps and the German word is Körper, both meaning "body."
A field with a finite number of members is known as a finite
field or Galois field.
Because the identity condition is generally required to 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 only 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.
), rational numbers (
), and real
numbers (
),
but not the integers (
), which form only a ring.
