U(Z[x]) ?

  1. Hi, I have to find the units of Z[x]. I am little unclear and my book does not go into detail. Does U(Z[x]) mean I need to find every polynomial with integer cooefficients that has a multiplicative inverse? or do I have to find the multiplicative identity? I was thinking about both cases and the number of polynomials with a multiplicative inverse is pretty limited, isn't it? f(x)=1 or f(x)=-1. As for inverses of polynomials, there would be none because if you multiply a polynomial with x by another polynomial with x then the powers of x get bigger.
     
  2. jcsd
  3. Hurkyl

    Hurkyl 16,090
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    It means to find all elements of Z[x] with a multiplicative inverse. (That's what it means to be a unit)

    That sounds plausible... can you work it into a rigorous proof?
     
  4. am I right about f(x)=1 and f(x)= -1 being the only polynomial in U(Z[x]) or is there more? would contradiction be the best way, how would I start it?

    tia
     
  5. Hurkyl

    Hurkyl 16,090
    Staff Emeritus
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    I generally hate answering this question. Learning how and when to be confident in your own work is important! But yes, you are correct.

    There are lots of ways. I would suggest starting with a literal translation of what you said:

    If p(x) and q(x) are nonconstant, then p(x)*q(x) is nonconstant.

    (Actually, you made a stronger statement, but I don't want to spoil figuring out how to translate that!)

    And see if you can prove this statement and relate it to inverses.
     
    Last edited: Oct 25, 2005
  6. ok thanks!
     
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