# Maximal Ideals of Q[x]

• September 28th 2008, 03:49 PM
roporte
Maximal Ideals of Q[x]
Find all maximal ideals of $Q[x]$ wich contains $\langle x^3+2x\rangle$.

Thanks!!
• September 28th 2008, 06:26 PM
NonCommAlg
Quote:

Originally Posted by roporte

Find all maximal ideals of $Q[x]$ wich contains $\langle x^3+2x\rangle$.

both ideals $I=, \ J=$ are maximal in $R=\mathbb{Q}[x]$ and $IJ=.$ if $K$ is a maximal ideal of $R$ and $IJ \subseteq K,$ then we have either $I \subseteq K$ or $J \subseteq K,$

because every maximal ideal is prime (more precisely, since $R$ here is a PID, maximality = primeness.) now since $I,J$ are maximal, we will either have $K=I$ or $K=J. \ \ \ \Box$