I am working through a Calculus book and I found an exercise, which I am not able to solve:
Let $K$ be a field of rational functions over $\mathbb{R}$ and let $a,b$ be arbitrary in $\mathbb{Q}$ but not all $b$ are $0$. And let $m,n$ be arbitrary in $\mathbb{N}$. We define $P:=\left.\left\{\frac{\sum_{i=0}^n a_ix^i}{\sum_{i=0}^m b_ix^i}\;\right|\;a_nb_m\lt 0\right\}.$
Is $P$ a prepositive cone?
Does anybody have a hint. My idea was to express $-1$ as a sum of squares but that didn't really work for me.