Maximal Ideals of Q[x]

**roporte** · September 28th 2008, 03:49 PM

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

Thanks!!

**NonCommAlg** · September 28th 2008, 06:26 PM

Originally Posted by roporte

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

both ideals $I=<x>, \ J=<x^2+2>$ are maximal in $R=\mathbb{Q}[x]$ and $IJ=<x^3+2x>.$ 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 $K=J. \ \ \ \Box$

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