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Math Help - list all maximal ideals

  1. #1
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    list all maximal ideals

    I seem to have difficulty with this type of problem - perhaps a lack of intuition about prime and maximal ideals. Here's the problem. I'm having trouble with part (d).

    For each of the following rings, list all maximal ideals.
    (a) \mathbb{Z}/90\mathbb{Z}
    (b) \mathbb{Q}[x]/(x^2+1)
    (c) \mathbb{C}[x]/(x^2+1)
    (d) \mathbb{Q}[x]/(x^3+x^2)

    My solutions for (a) through (c):

    (a) These are generated by the prime factors of 90: (2), (3), and (5).
    (b) Since x^2+1 is irreducible in \mathbb{Q}[x], this is a field, so the only maximal ideal is (0).
    (c) \mathbb{C}[x]/(x^2+1) = \mathbb{C}[x]/((x+i)(x-i)) \cong \mathbb{C}[x]/(x+i)\times\mathbb{C}[x]/(x-i) \cong \mathbb{C}\times\mathbb{C}, so the maximal ideals are ((0,1)) and ((1,0)), which correspond to the ideals (x+i) and (x-i) in the original ring.

    I think that answer is correct, but I'm not entirely clear on how I got from the ideals in \mathbb{C}\times\mathbb{C} to the ideals in the original ring.

    For (d), I think it's true that \mathbb{Q}[x]/(x^3+x^2) \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x+1), and the second is isomorphic to \mathbb{Q}. But what are the maximal ideals of \mathbb{Q}[x]/(x^2)?

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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: list all maximal ideals

    The key to all of these is the lattice (fourth) isomorphism theorem. Namely, you know that the maximal ideals of a quotient ring R/I are the maximal ideals of the form M/I for M a maximal ideal of R containing I. So, let's see here:

    a) Yes, as you pointed out, the maximal ideals are the principal ones generated by 2,3,5. This follows from what I said since the maximal ideals of \mathbb{Z} that contain (90) are the principal ideals of the form (p) where p\mid 90 is prime.

    b) Exactly. Since \mathbb{Q} is a field, you know that \mathbb{Q}[x] is a PID, and since x^2+1 is irreducible we must have that (x^2+1) is prime, and thus maximal. Thus, the only maximal ideals containing (x^2+1) are the ideal itself!

    c) Exactly. Since \mathbb{C} is algebraically closed, you know that all the maximal ideals are the principal ones with linear generators. Since the only linear factors dividing x^2+1 are x\pm i your answer is correct.

    d) Once again, since \mathbb{Q}[x] is a PID, the maximal ideals containing (x^3+x^2) are the principal ideals generated by the irreducible factors of x^3+x^2. In other words, the maximal ideals above (x^3+x^2) are the ideals (x+1) and (x). By the way, using this idea, you see that the maximal ideals of \mathbb{Q}[x]/(x^2) is just the ideal (x).

    Best,

    Alex
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