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Math Help - Ring

  1. #1
    Senior Member slevvio's Avatar
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    Ring

    Show that the only units of  \mathbb{C} [ X] are the non-zero complex number, i.e. The non zero complex polynomials.

    Hint: suppose p =  \sum_{r=o}^{n} a_r X^r is a unit and let  p^{-1} = \sum_{s=o}^{m} b_s X^s be its inverse.
    Consider the highest power of X in the product  p p^{-1} .

    Can anybody give me a hand with this, any help would be much appreciated. I can't think why considering the highest power of x would help me
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  2. #2
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    Quote Originally Posted by slevvio View Post
    Show that the only units of  \mathbb{C} [ X] are the non-zero complex number, i.e. The non zero complex polynomials.

    Hint: suppose p =  \sum_{r=o}^{n} a_r X^r is a unit and let  p^{-1} = \sum_{s=o}^{m} b_s X^s be its inverse.
    Consider the highest power of X in the product  p p^{-1} .

    Can anybody give me a hand with this, any help would be much appreciated. I can't think why considering the highest power of x would help me

    Suppose deg(p)=n\,\Longrightarrow\,deg(pq)\geq n\,\,\,\forall\,q\in\mathbb{C}[x]\,\Longrightarrow\,p is a unit iff n=0=deg(1)

    Tonio
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  3. #3
    Senior Member slevvio's Avatar
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    Thanks Tonio I see this now

    clearly  pp^{-1} = X^{n+m} + l.o.t. = 1

    so degree (1) = 0 = degree (  X^{n+m} ) = n+m

    so n = - m but n, m both greater than or equal to 0 , hence n = m = 0 and the power series of p has only one term, a complex number. Thanks
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