Field Axioms

The field axioms are generally written in additive and multiplicative pairs.

Name	Addition	Multiplication
Commutativity	$a+b = b+a$	$ab = ba$
Associativity	$(a+b)+c = a+(b+c)$	$(ab)c = a(bc)$
Distributivity	$a(b+c) = ab+ac$	$(a+b)c = ac+bc$
Identity	$a+0 = a = 0+a$	$a\cdot 1 = a = 1\cdot a$
Inverses	$a+(-a) = 0 = (-a)+a$	$aa^{-1} = 1 = a^{-1}a\hbox{ if }a \not = 0$

See also Algebra, Field