Find all maximal ideals of $\displaystyle Q[x]$ wich contains $\displaystyle \langle x^3+2x\rangle$.
Thanks!!
both ideals $\displaystyle I=<x>, \ J=<x^2+2>$ are maximal in $\displaystyle R=\mathbb{Q}[x]$ and $\displaystyle IJ=<x^3+2x>.$ if $\displaystyle K$ is a maximal ideal of $\displaystyle R$ and $\displaystyle IJ \subseteq K,$ then we have either $\displaystyle I \subseteq K$ or $\displaystyle J \subseteq K,$
because every maximal ideal is prime (more precisely, since $\displaystyle R$ here is a PID, maximality = primeness.) now since $\displaystyle I,J$ are maximal, we will either have $\displaystyle K=I$ or $\displaystyle K=J. \ \ \ \Box$