Calculating the class group problem

In summary, the conversation revolves around calculating the class group of a cubic equation. The ring of integers is generated by {1,alpha,alpha^2} where alpha is a root of the equation. The factorization of 2*O_K and 3*O_K is found, with the minkowski bound being 3. The class group is generated by at most two ideals, but proving that these ideals are not principal is difficult due to the complicated norm. However, by calculating the norm of the ideals and using the fact that the norm of a principal ideal must divide the norm of the ring, it is shown that the class group is trivial.
  • #1
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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  • #2
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.
 

Related to Calculating the class group problem

What is the class group problem?

The class group problem is a mathematical problem that involves determining the structure of the class group of a number field. The class group is a group that captures the arithmetic properties of the field, such as the existence of unique factorization.

Why is calculating the class group important?

Calculating the class group is important because it allows us to study the arithmetic properties of number fields, which have applications in algebra, number theory, and cryptography. It also helps us understand the behavior of prime numbers and the distribution of solutions to certain equations.

What is the class number of a number field?

The class number of a number field is the order of its class group. It represents the number of different types of ideals in the field, which can have important consequences in number theory.

How is the class group problem solved?

The class group problem is typically solved using various algorithms and computational methods, such as the Lenstra-Lenstra-Lovász algorithm and the elliptic curve method. These methods involve manipulating and analyzing the properties of the number field and its associated objects, such as ideal classes and class groups.

What are some applications of the class group problem?

The class group problem has applications in various fields of mathematics, including number theory, algebraic geometry, and cryptography. It has also been used to study the behavior of prime numbers and to solve certain mathematical puzzles and problems.

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