I know that in general if $R[u]$ is the ring obtained by adjoining an element $u$ to a ring $R$, then $R[u]\cong R[x]/I$ for some ideal $I$ such that $I\cap R=\{0\}$.
In a particular instance, I'm working with $u=\sqrt{2}+\sqrt{3}$, so I'm wondering if there is a concrete description of an ideal $I$, say generated by some polynomials of $\mathbb{Q}[x]$ or something like that, such that $\mathbb{Q}[x]/I\cong\mathbb{Q}[u]$.
It's clear there is a surjective homomorphism from $\mathbb{Q}[x]\to\mathbb{Q}[u]$, so I would take the kernel, which is just the set of all polynomials with $u$ a root. Is this ideal generated by anything easy to write down? Or is that the best description I can give?