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Math Help - Max Ideals of Z3[x]

  1. #1
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    Max Ideals of Z3[x]

    Claim: The set of all maximal ideals in \mathbb{Z}_3[x] is not in one-to-one correspondence with \mathbb{Z}_3 = \{0,1,2\}.

    I haven't found a proof yet. Seems that I have to show that the set of all maximal ideals in \mathbb{Z}_3[x] is is less than 3 or greater than 3.

    Ideas:

    I can show that (1+x^2) is a maximal ideal using 1st Iso. Thm. and the fact that \mathbb{Z}_3 / (1+x^2) is a field. I could probably show that (1+x) is a max ideal similarly; haven't done this yet as I'm not sure it helps. It might be true that these are the only maximal ideals of \mathbb{Z}_3[x] but I'm not sure how to prove it. If these are true, then we're done.
    Last edited by huram2215; May 19th 2010 at 03:26 PM. Reason: Error on 1st entry
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  2. #2
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    Quote Originally Posted by huram2215 View Post
    Claim: The set of all maximal ideals in \mathbb{Z}_3[x] is not in one-to-one correspondence with \mathbb{Z}_3 = \{0,1,2\}.

    I haven't found a proof yet. Seems that I have to show that the set of all maximal ideals in \mathbb{Z}_3[x] is is less than 3 or greater than 3.

    Ideas:

    I can show that (1+x^2) is a maximal ideal using 1st Iso. Thm. and the fact that \mathbb{Z}_3 / (1+x^2) is a field. I could probably show that (1+x) is a max ideal similarly; haven't done this yet as I'm not sure it helps. It might be true that these are the only maximal ideals of \mathbb{Z}_3[x] but I'm not sure how to prove it. If these are true, then we're done.
    recall that if F is a field, then an ideal I of F[x] is maximal iff I=\langle f(x) \rangle, where f(x) \in F[x] is irreducible. so all the following ideals are maximal in \mathbb{Z}_3[x]:

    \langle x \rangle, \ \langle x+1 \rangle, \ \langle x+2 \rangle, \ \langle x^2+1 \rangle, \ \langle x^2+x+2 \rangle, \ \langle x^2+2x+2 \rangle, \dots
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