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Thread: Units

  1. #1
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    Units

    Show that $\displaystyle \left( \mathbb{Z}_{[x]} \right)^* = \{-1,1\} $
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Suppose $\displaystyle a_n x^n + \cdots + a_1 x + a_0 \in \mathbb{Z} [x] $ is a unit.

    Then there exists $\displaystyle p(x) $ s.t. $\displaystyle (a_n x^n + \cdots + a_1 x + a_0)p(x) = 1 $. It is evident $\displaystyle p(x) \neq 0 $.

    $\displaystyle (a_n p(x)) x^n + \cdots + (a_1 p(x)) x + a_0 p(x) = 1 $

    Therefore $\displaystyle a_i p(x) = 0 \; 1 \leq i \leq n $ and $\displaystyle a_0 p(x) = 1 $.

    Since $\displaystyle \mathbb{Z} [x] $ is an integral domain and $\displaystyle p(x) \neq 0 $, $\displaystyle a_i = 0 \; 1 \leq i \leq n $.

    The only way for $\displaystyle a_0 p(x) = 1 $ is if $\displaystyle p(x) = c $, where $\displaystyle c \in \mathbb{Z} $. So $\displaystyle a_0 c = 1 \Longleftrightarrow a_0 = 1 , \; c = 1 $ or $\displaystyle a_0 = -1 , \; c = -1 $

    Hence $\displaystyle a_n x^n + \cdots + a_1 x + a_0 = -1 $ or $\displaystyle 1 $. Which means $\displaystyle \left( \mathbb{Z} [x] \right)^* = \{-1,1\} $.
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  3. #3
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    Quote Originally Posted by chiph588@ View Post

    Therefore $\displaystyle a_i p(x) = 0 \; 1 \leq i \leq n $ and $\displaystyle a_0 p(x) = 1 $.
    i don't think you can make this conclusion!

    well, if $\displaystyle (a_nx^n + \cdots + a_0)(b_mx^m + \cdots + b_0)=1, \ a_n \neq 0, \ b_m \neq 0,$ and n > 0 or m > 0, then we must have $\displaystyle a_nb_m=0,$ which is not possible in an integral domain. so n = m = 0, and the rest is

    obvious. this also shows that, in general, for any integral domain $\displaystyle R,$ the set of units of $\displaystyle R[x]$ is exactly equal to the set of units of $\displaystyle R.$

    Remark: there's a well-known result that in any commutative ring with unity $\displaystyle R,$ a polynomial $\displaystyle \sum_{k=0}^n a_kx^k \in R[x]$ is a unit iff $\displaystyle a_0$ is a unit of $\displaystyle R$ and all $\displaystyle a_k, \ k \neq 0,$ are nilpotent.
    Last edited by NonCommAlg; Apr 22nd 2009 at 12:44 AM. Reason: added the remark!
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