= Ordered field
{title2=$(K,\leq)$}
An ordered field is a <field> $K$ with a <total order> compatible with its arithmetic: $a\le b$ implies $a+c\le b+c$, and $0\le a,0\le b$ imply $0\le ab$. The <rational numbers> and <real numbers> have their usual ordered-field structures.
Every nonzero square is positive. Indeed, either $x>0$ or $-x>0$, and multiplying the positive element by itself gives $x^2>0$. In particular $1>0$, so every sum $1+\cdots+1$ is nonzero and an ordered field has <characteristic zero>. The <complex numbers> cannot admit such an order because $i^2=-1$ would force $-1>0$.
An ordered field need not satisfy <order completeness> or the <Archimedean property>. In the rational-function <field> $\mathbb R(t)$, declaring a nonzero <rational function> positive when it is positive for all sufficiently large real $t$ defines an order in which $t$ exceeds every real constant. Order compatibility and completeness are separate assumptions.
Back to article page
