ordered field and R can embed

Question

errr.. Is there exists any field $\mathbb{F} \neq \mathbb{R} $ such that:

• ordered field

• $\mathbb{R} \hookrightarrow \mathbb{F}$

Answer 1

Consider the field of rational functions $\mathbb{R}(x)$, in which an element $\frac{a_nx^n +\cdots+ a_0}{b_m x^m + \cdots + b_0}$ is considered a positive element if and only if $\frac{a_n}{b_m}$ is positive as a real number. You can verify that this satisfies all the properties of an ordered field, and $\mathbb{R}$ sits naturally inside it as a subfield.