field (mathematics)

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.All fields are rings, but not conversely. Fields differ from rings most importantly in the requirement that division be possible, but also, in modern definitions, by the requirement that the multiplication operation in a field be commutative. Otherwise the structure is a so-called skew field (better known as a division ring), although historically division rings were called fields and fields were commutative fields.
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