Fields [and axioms]

With ⊕ being + and ⊗ being × rational, real and complex numbers are all examples of fields. Ordinary integers lack inverses under ×, but reduction modulo a prime p gives a finite field. Since the 1700s many examples of fields have arisen, particularly in algebra and number theory. The general axioms for fields as given here emerged around the end of the 1800s. Shorter versions can undoubtedly be found. (See page 1168.)