Calculating the class group problem

[hippos]

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

 

[lippyka]

ymbol's answer:It looks like you need to calculate the norm of the two ideals <3,alpha-1> and <3,alpha^2+alpha+2>. The norm is defined as the product of all the prime ideals that appear in the factorization. In this case, the norm of <3,alpha-1> is 3 and the norm of <3,alpha^2+alpha+2> is 3^2=9. Now you can use the fact that the norm of a principal ideal must divide the norm of the ring (which is 2*O_K). Since the norm of <3,alpha-1> and <3,alpha^2+alpha+2> are both 3 and the norm of 2*O_K is 2, this implies that both <3,alpha-1> and <3,alpha^2+alpha+2> are principal ideals. Therefore, the class group of f=x^3-5x-5 is trivial.