Fields

Why

We generalize the algebraic structure of addition and multiplication over the rationals.

Definition

A field is a ring $(R, +, \cdot )$ for which $\cdot $ is commutative (i.e., $ab = ba$ for all $a, b \in R$) and $\cdot $ has inverses for all elements except $0$. In this case, we refer to field addition and field multiplication.

Notation

Since our guiding example is the set of rationals $\Q $ with addition and multiplication defined in the usual manner, and we use a bold font for $\Q $, we tend to denote an arbitrary field by $\F $, a mnemonic for “field.”

Field operations

Along with field addition and field multiplication, we call the function which takes an element of a field to its additive inverse and the function which takes an element of a field to its multiplicative inverse the field operations.