Linear Algebra, 5th Edition

Appendix C Fields

The set of real numbers is an example of an algebraic structure called a field. Basically, a field is a set in which four operations (called addition, multiplication, subtraction, and division) can be defined so that, with the exception of division by zero, the sum, product, difference, and quotient of any two elements in the set is an element of the set. More precisely, a field is defined as follows.

Definitions.

A field F is a set on which two operations $+$ and $\cdot$ (called addition and multiplication, respectively) are defined so that, for each pair of elements x, y in F, there are unique elements in F, denoted $x+y$ and $x\cdot y$, and such that the following conditions hold for all elements a, b, c in F.

1. (F 1) $a+b=b+a$ and $a\cdot b=b\cdot a$