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Greatest common divisor of polynomials

  1. Jul 7, 2009 #1
    HI there,

    I have a tiny question concerning the gcd of polynomials. Assume, [itex]\chi[/itex] is the greatest common divisor of the polynomails [itex]p_{ij}, i,j=1,2[/itex]. I then form
    [tex]
    q_{11}=p_{11}^2+p_{12}^2,\quadd q_{12}=p_{11}p_{21}+p_{12}p_{22},\quadd q_{21}=p_{21}p_{11}+p_{22}p_{12},\quadd q_{22}=p_{21}^2+p_{22}^2
    [/tex]

    which is in fact a matrix multiplication:
    [tex]
    \left(\begin{array}{cc}
    q_{11} & q_{12} \\ q_{21} & q_{22}
    \end{array}\right) = \left(\begin{array}{cc}
    p_{11} & p_{12} \\ p_{21} & p_{22}
    \end{array}\right)
    \left(\begin{array}{cc}
    p_{11} & p_{12} \\ p_{21} & p_{22}
    \end{array}\right)^T
    [/tex].

    I think everyone would agree that [tex]\chi^2[/itex] is a common divisor of all the q's. But is it also the greatest one...? I couldn't find a counterexample and couldn't prove it either.

    Thanks a lot :smile:
     
    Last edited: Jul 7, 2009
  2. jcsd
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