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hippos

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**Stuck calculating the class group of a cubic**

I'm trying to calculate the class group of f=x^3-5x-5. I know that the ring of integers (O_K) is generated by {1,alpha,alpha^2} where alpha is a root of f, and I've found the factorization of 2*O_K, and 3*O_K (the minkowski bound is 3).

2*O_K=<2> (2 is inert)

3*O_K=<3,alpha-1><3,alpha^2+alpha+2>

So the class group is generated by (at most) <3,alpha-1>and <3,alpha^2+alpha+2>. My problem is that the norm is really ugly, so I'm having trouble proving that an ideal isn't principle. Any thoughts would be appreciated.

hippos

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