Field 2001-10-19 00:26:16 2008-05-08 15:44:01 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A \emph{field} is a set $F$ together with two binary operations on $F$, called addition and multiplication, and denoted $+$ and $\cdot$, satisfying the following properties, for all $a,b,c \in F$: \begin{enumerate} \item $a + (b+c) = (a+b)+ c$ (associativity of addition) \item $a+b = b+a$ (commutativity of addition) \item $a+0 = a$ for some element $0 \in F$ (existence of zero element) \item $a+(-a) = 0$ for some element $-a \in F$ (existence of additive inverses) \item $a\cdot (b\cdot c) = (a\cdot b)\cdot c$ (associativity of multiplication) \item $a\cdot b = b\cdot a$ (commutativity of multiplication) \item $a\cdot 1 = a$ for some element $1 \in F$, with $1 \neq 0$ (existence of unity element) \item If $a \neq 0$, then $a \cdot a^{-1} = 1$ for some element $a^{-1} \in F$ (existence of multiplicative inverses) \item $a\cdot (b+c) = (a\cdot b) + (a\cdot c)$ (distributive property) \end{enumerate} Equivalently, a field is a commutative ring $F$ with identity such that: \begin{itemize} \item $1 \neq 0$ \item If $a \in F$, and $a \neq 0$, then there exists $b \in F$ with $a \cdot b = 1$. \end{itemize}