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Math Help - I don't really understand polynomial quotient rings

  1. #1
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    I don't really understand polynomial quotient rings

    For example, what is the difference between

    \mathbb{Z}_2[x]/(x^2 + 1) and \mathbb{Z}_2[x]/(x^2 + x + 1)?
    What do they look like?

    Is \mathbb{Z}_2[x]/(x^2 + 1) \cong \mathbb{F}_2[x]/(x^2 + 1)?

    I'm having trouble "getting it" if that makes sense.
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  2. #2
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    Quote Originally Posted by electricjaguar View Post
    For example, what is the difference between

    \mathbb{Z}_2[x]/(x^2 + 1) and \mathbb{Z}_2[x]/(x^2 + x + 1)?
    What do they look like?

    Is \mathbb{Z}_2[x]/(x^2 + 1) \cong \mathbb{F}_2[x]/(x^2 + 1)?

    I'm having trouble "getting it" if that makes sense.
    \mathbb{Z}_2[x]/(x^2 + 1) = \{ a+bx+x^2 + 1|a,b \in \mathbb{Z}_2 \}

    \mathbb{Z}_2[x]/(x^2 + x + 1)= \{ a+bx+x^2 +x+ 1|a,b \in \mathbb{Z}_2 \}

    make sure the polynomials are irreducible. also, the fields have the same cardinality [which is 4].

    Yes b/c F_2[x]=\mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}_2
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  3. #3
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    Thank you, I have more questions that I'm just going to put in this thread so as not to cloud up the forum.

    Is it true that \mathbb{Z}/(16) contains a unique maximal ideal, but \mathbb{Z}/(12) does not? I think this is the case by the correspondence theorem


    Also, I can see why \mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}, but why is \mathbb{R}[x]/(x^2 + x + 1) \cong\mathbb{C}?
    Last edited by mr fantastic; March 18th 2009 at 01:21 AM. Reason: Merged posts
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  4. #4
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    I also have a question about \mathbb{Z}_2[x]/(x^2 + 1). Isn't one of its elements x^2 + 1 + 1 = x^2 + 2 = x^2 which has a root in \mathbb{Z}_2, namely, x = 0? Doesn't this show that this element is reducible and hence the quotient is not a field?

    edit: nevermind
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  5. #5
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    Quote Originally Posted by electricjaguar View Post
    Thank you, I have more questions that I'm just going to put in this thread so as not to cloud up the forum.

    Is it true that \mathbb{Z}/(16) contains a unique maximal ideal, but \mathbb{Z}/(12) does not? I think this is the case by the correspondence theorem


    Also, I can see why \mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}, but why is \mathbb{R}[x]/(x^2 + x + 1) \cong\mathbb{C}?
    yes b/c  8\mathbb{Z}/16\mathbb{Z} \subseteq 4\mathbb{Z}/16\mathbb{Z} \subseteq 2\mathbb{Z}/16\mathbb{Z}
    so 2\mathbb{Z}/16\mathbb{Z} is uniquely maximal

    but

    6\mathbb{Z}/12\mathbb{Z} \not \subseteq 4\mathbb{Z}/12\mathbb{Z} \not \subseteq 3\mathbb{Z}/12\mathbb{Z} <br />
\not \subseteq 2\mathbb{Z}/12\mathbb{Z}
    ie there is not a maximal ideal that contains all those ideals
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