Example

— 1) Every ordered field $K$ is an ordered extension of $\mathbf{Q}$. Indeed $K$ is an extension of $\mathbf{Q}$, since it is of characteristic zero, and on the other hand $\mathbf{Q}$ can only be ordered in one way, as we have just seen.

2) Let $K$ be an ordered field, and let $K(X)$ be the field of rational functions in one indeterminate over $K$. Let us define an ordering on the polynomial ring $K[X]$ by taking the positive elements to be $0$ and those polynomials whose leading coefficient is positive. In this way we obtain a totally ordered ring whose ordering extends that of $K$. By applying Prop. 2 we give $K(X)$ the structure of an ordered extension of $K$. *For $K = \mathbf{R}$ it can be shown that the order relation defined on $K(X)$ in this way is that of growth near $+\infty$ (cf. VI, p. 24, Prop. 4).*