Let $f(x)$ a polynomial in $\mathbb Z[\frac{1+\sqrt n}{2}][x]$ ($n=4k+1$). I wish to find to a polynomial $g$ in $\mathbb Z[\frac{1+\sqrt n}{2}][x]$ such that $fg\in \mathbb Z[x]$.
One possible approach is setting $g$ to be $f$ with all $\sqrt n$ in the coefficients replaced by $-\sqrt n$. I don't know how to prove that the resulting $fg\in \mathbb Z[x]$.
I think $n=4k+1$ may be a necessary hypothesis, see here for an example.