1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Euclidean rings
  1. Another Euclidean ring (Replies: 9)

  2. Euclidean alg (Replies: 8)

  3. Euclidean space (Replies: 1)