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Norm of an Ideal

  1. Mar 25, 2012 #1
    I need to calculate the norm of the ideal

    p = (3, 1 - √-5)

    All the information I have is that it's a prime ideal.

    I managed to calculate the normal of the ideal q = (3, 1 + √-5) (which was 3) by finding a the determinant of a base change matrix by considering an integral basis

    Here I'm not sure how to do that (in the other example I managed to show an equivelence relation so that I could find an integral bases)

    Here is a similar example with the ideal p1 = (2, 1 + √-17)

    34sflaf.png

    Any help appreciated, thanks.
     
  2. jcsd
  3. Mar 25, 2012 #2

    Hurkyl

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    Gold Member

    Well, you seem to have been shown two methods: compute a determinant, and compute a residue ring. What difficulty have you had trying to use either method?


    P.S. when computing the residue ring, I often find it easier to think of your ring as being the quotient of a polynomial ring:

    [tex]\begin{align}
    \mathbb{Z}[\sqrt{-17}] / (2, 1 + \sqrt{-17}) &\cong
    \left( \mathbb{Z}[x] / (x^2 + 17) \right) / (2, 1 + x)
    \\ &\cong \mathbb{Z}[x] / (2, 1+x, x^2 + 17)
    \\ &\cong \left(\mathbb{Z}[x] / (2)\right) / (1+x, x^2 + 17)
    \\ &\cong \cdots\end{align}[/tex]
     
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