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Euclidean rings

  1. Nov 21, 2006 #1
    [tex]\displaystyle{\zeta = e^{{2\pi i} \over 5}}[/tex]
    I need to show that [itex]Z[\zeta][/itex] is a Euclidean ring.

    The only useful technique I know about is showing that given an element [itex]\epsilon \in Q(\zeta)[/itex] we can always find [itex]\beta \in Z[\zeta][/itex] such that [itex]N(\epsilon - \beta) < 1[/itex] (using the standard norm for the euclidean function).

    This usually involves finding a general expression for the norm and then saying that you can choose beta such that the difference of each basis element is less than 1/2, and then showing that this means you can also get the norm less than 1.

    However, the expression I got for the norm here didn't seem to lend itself to this method.

    Any suggestions on how to do this?
    Last edited: Nov 21, 2006
  2. jcsd
  3. Nov 21, 2006 #2

    matt grime

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    show it has a thingummmy - euclidean function, can't remember the precise name, that might help.
  4. Nov 21, 2006 #3
    Of course, but that's the point. The problem is I can't show the Norm is less than one if the coefficients are less than 1/2 and don't know any other techniques.
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