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Ring problem

  1. Jul 26, 2009 #1
    I am having a problem with some abstract algebra and I was wondering if anyone could help and give me some insight. The problem is as follows:

    Give an explanation for your answer, long proof not needed:
    Determine U(Z[x])
    Determine U(R[x])

    These are in regards to rings. I know for U(Z[x]) it is something like f(x)=1 g(x)=-1 but I don't know why.

    As for U(R[x]) I am rather stuck. Any help or nudging in the right direction would be greatly appreciated.
     
  2. jcsd
  3. Jul 26, 2009 #2
    For the first case suppose [tex]f \in \mathbb{Z}[x][/tex] is a unit. Then there exists another element [tex]g \in \mathbb{Z}[x][/tex] such that fg=1. If deg f > 0 what can you say about the degree of fg? Can fg be the multiplicative identity when deg fg >0? If deg f = 0 what can we say about f and g?

    Similar considerations suffice for the ring of polynomials with real coefficients.
     
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