Field Axioms

Definition

Property	Premise	Addition	Multiplication
Commutative	$\forall x,y\in \mathbb{R}$	$x+y=y+x$	$x\cdot y=y\cdot x$
Associative	$\forall x,y,z\in \mathbb{R}$	$(x+y)+z=x+(y+z)$	$(x\cdot y)\cdot z=x\cdot (y\cdot z)$
Identity Element	$\exists 0,1\in \mathbb{R},0\ne 1\ni \forall x\in \mathbb{R}$	$x+0=x=0+x$	$x\cdot 1=x=x\cdot 1$
Inverse Element	$\forall x,y\in \mathbb{R},y\ne 0\exists (-x),y^{-1}\in \mathbb{R}$	$x+(-x)=0=(-x)+x$	$y\cdot y^{-1}=1=y^{-1}\cdot y$
Distributive	$\forall x,y,z\in \mathbb{R}$	$x(y+z)=xy+xz=(y+z)x$

==All familiar techniques of algebra can be derived from those 9 properties.==