Examples

1) The field $\mathbf{Q}$ of rational numbers is an ordered field. 2) A subfield of an ordered field, with the induced ordering, is an ordered field. 3) \ The field of real numbers is an ordered field. \*

Let $K$ be an ordered field. For all $x \in K$ we put

Then we have $\operatorname{sgn}(xy) = \operatorname{sgn}(x) \operatorname{sgn}(y)$; we call $\operatorname{sgn}(x)$ the sign of $x$. The map $x \mapsto \operatorname{sgn}(x)$ from $K^*$ to the multiplicative group $\{-1, +1\}$ is a surjective homomorphism whose kernel, the set of strictly positive elements of $K$, is a subgroup of $K^*$ of index 2.

Conversely, if $K$ is a commutative field and $s : K^* \to \{-1, +1\}$ is a surjective homomorphism whose kernel is closed under addition, then $s$ is the sign map for a unique ordered field structure, where the set of strictly positive elements is the kernel of $s$.

For all $x$ and $y$ in $K$ we have $x = \operatorname{sgn}(x)|x|$ and $|xy| = |x||y|$.

On the other hand every ordered field is of characteristic zero (Prop. 1).