Field (mathematics)

In mathematics, a field is a certain kind of algebraic structure. In a field, one can add (formula_1), subtract (formula_2), multiply (formula_3) and divide (formula_4) two numbers (with division only possible if formula_5 is non-zero). A field is a special ring in which division is possible. Both the set of rational numbers and the set of real numbers are examples of fields. One can also use the symbol formula_6 as a variable for a field.
Rules.
Since a field is always a ring, it consists of a set (represented here with the letter R) with two operations: addition (+) and multiplication (•). The properties of a ring are:
A field is a ring (so it follows the rules above), with these extra rules:
Another way of describing a field is this:
Examples of fields.
Examples of fields include: 
In fact, among the subsets of complex numbers, the largest field is formula_9 and the smallest field is formula_7. The set of integers, formula_12, is not a field, because the result of a division is not always an integer.