Find in $\mathbb{Q}$[x] a subset $\mathbf{N}$ so that axioms P(i) and p(ii) are both satisfied.
I have this definition for axiom P from the integers:
$\mathbb{Z}$ contains a non-empty subset $\mathbf{N}$ such that:
(i) each element of $\mathbb{Z}$ belongs to exactly one of the sets $\mathbf{N}$, $\{0\}$, $-\mathbf{N}$ where $-\mathbf{N}$ denotes the set $\{-x : x\in \mathbf{N}\}$
(ii) for all $a,b$ $\in \mathbf{N}$ we have $a$ $\oplus$ b $\in$ $\mathbf{N}$ and $a$ $\bullet$ $b$ $\in$ $\mathbf{N}$
I'm pretty sure that this axiom also holds for $\mathbb{Q}$[x], however I am struggling to identify the set $\mathbf{N}$ for which it holds.
This set is the set of polynomials of positive leading coefficient.