Marcus Number fields exercise 17 chapter 4

Question

I know this question has already been posted, but I don't manage to understand the comments.

Exercise $17$ $(e)$ on Marcus' Number Fields, Chapter $4$

My problem is exactly the same as the one the other user has pointed out: I managed to show that $|E|T \subset S+U$ , but I have no idea how to proceed to conclude.

Answer 1

$U$ is a maximal ideal so $S + U$ is either $U$ or the whole ring $T$. $S + U \not\subseteq U$ (take any $s \not\in Q$), so $S + U = T$.