Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers. A field is thus a fundamental algebraic structure.

An archaic name for a field is rational domain. The French term for a field is corps and the German word is K├Ârper, 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 ($\mathbb{C}$), rational numbers ($\mathbb{Q}$), and real numbers ($\mathbb{R}$), but not the integers ($\mathbb{Z}$), 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.