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Prime ideal in ring

  1. May 8, 2007 #1
    1. The problem statement, all variables and given/known data

    Take the ideal

    [tex] I = < 6, 3 + 3 \sqrt{-17} >[/tex]

    in the ring [itex]Z [ \sqrt{-17} ][/itex].

    Determine whether this ideal is prime or not.

    2. Relevant equations

    [tex]<18> = I^2 [/tex]

    There is no element [itex]\alpha \in Z [ \sqrt{-17} ] [/itex] such that [itex] 18 = \alpha^2[/itex]

    3. The attempt at a solution

    I really don't know how to go about doing this. I have the definition of a prime ideal P.

    [itex] P [/itex] is a prime ideal [itex] \Leftrightarrow [/itex] if [itex] ab \in P[/itex] then [itex] a \in P [/itex] or [itex] b \in P [/itex].

    And I see that [itex] < 6, 3 + 3 \sqrt{-17} > = 6 Z [ \sqrt{-17} ] +( 3 + 3 \sqrt{-17}) Z [ \sqrt{-17} ][/itex].

    Is it also possible to pull out the [itex] Z [ \sqrt{-17} ][/itex] from the above equation?

    Thanks
     
    Last edited: May 8, 2007
  2. jcsd
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