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Daniel Miller
Daniel Miller
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The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question:

Is there a quantitative statement relating the class number of a number field to the failure of unique factorization in the maximal order - other than h = 1$h = 1$ iff R$R$ is a ufd?

In what sense does a maximal order of class number 3 "fail more" to be a ufd than a maximal order of class number 2?

Is it true that an integer in a field of greater class number will have more distinct representations as the product of irreducible elements than an integer in a field with smaller class number?

The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question:

Is there a quantitative statement relating the class number of a number field to the failure of unique factorization in the maximal order - other than h = 1 iff R is a ufd?

In what sense does a maximal order of class number 3 "fail more" to be a ufd than a maximal order of class number 2?

Is it true that an integer in a field of greater class number will have more distinct representations as the product of irreducible elements than an integer in a field with smaller class number?

The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question:

Is there a quantitative statement relating the class number of a number field to the failure of unique factorization in the maximal order - other than $h = 1$ iff $R$ is a ufd?

In what sense does a maximal order of class number 3 "fail more" to be a ufd than a maximal order of class number 2?

Is it true that an integer in a field of greater class number will have more distinct representations as the product of irreducible elements than an integer in a field with smaller class number?

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Dror Speiser
Dror Speiser
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Class number measuring the failure of unique factorization

The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question:

Is there a quantitative statement relating the class number of a number field to the failure of unique factorization in the maximal order - other than h = 1 iff R is a ufd?

In what sense does a maximal order of class number 3 "fail more" to be a ufd than a maximal order of class number 2?

Is it true that an integer in a field of greater class number will have more distinct representations as the product of irreducible elements than an integer in a field with smaller class number?