Thread: Maximal Ideals of Q[x]
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Old September 28th, 2008, 08:26 PM
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Originally Posted by roporte View Post

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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