I'm working on a question that asks to show that the field of fractions of the polynomial ring with real coefficients is an ordered field by defining $$\frac{p}{q} > 0 \iff \frac{a_m}{b_n} > 0 $$ where $p$ and $q$ are polynomials, and $a_m$ and $b_n$ are the leading coefficients of $p$ and $q$ respectively.
I can't seem to figure out how to wrangle a total ordering of this field from the above definition. It seems like defining a comparable based on the leading coefficient would lead to all sorts of inconsistencies; for example, $$ \frac{x^3}{x^2} = \frac{1}{1} = 1 = \frac{x}{x^2}$$ if ordering were determined solely by the magnitude of $\frac{a_m}{b_n}$. Any help would be much appreciated!
This definition must be taken together with the general rule for ordered fields that $x>y$ if and only if $x-y>0$. In order to compare two rational functions we cannot just compute $a_m/b_n$ for each of them -- we need to actually compute their difference (by putting on a common denominator and so forth) and then apply the rule.
Interpreted in this way, the definition is equivalent to declaring that we compare two rational functions $f$ and $g$ by
Clearly this is a transitive relation and irreflexive relation, and it is also total, because the rational function $f-g$ is either zero or eventually positive or eventually negative -- it can't keep wiggling around the axis forever because its numerator has only finitely many roots.
Note that the definition does not say that $\frac{x^3}{x^2}$, $\frac11$, and $\frac{x}{x^2}$ are the same -- only that they have the same sign. And indeed they all count as positive.