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Thread: Max Ideals of Z3[x]

  1. #1
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    Max Ideals of Z3[x]

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

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

    Ideas:

    I can show that $\displaystyle (1+x^2)$ is a maximal ideal using 1st Iso. Thm. and the fact that $\displaystyle \mathbb{Z}_3 / (1+x^2)$ is a field. I could probably show that $\displaystyle (1+x)$ 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 $\displaystyle \mathbb{Z}_3[x]$ 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 02:26 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 $\displaystyle \mathbb{Z}_3[x]$ is not in one-to-one correspondence with $\displaystyle \mathbb{Z}_3 = \{0,1,2\}$.

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

    Ideas:

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

    $\displaystyle \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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