osztályszámok

osztályszámok, often translated as "class numbers," are a fundamental concept in abstract algebra, particularly in the study of number fields. They are an invariant associated with an algebraic number field that measures how far its ring of integers is from being a unique factorization domain (UFD). In simpler terms, class numbers help us understand if and how unique factorization fails in the ring of integers of a number field.

The ring of integers of a number field $K$, denoted by $\mathcal{O}_K$, is the set of algebraic integers that are elements of $K$. A UFD is a ring where every non-zero, non-unit element can be uniquely factored into a product of irreducible elements (up to order and units). While the integers $\mathbb{Z}$ are a UFD, this property does not generally hold for the rings of integers of number fields.

The class number of a number field $K$ is precisely the order of its ideal class group, denoted by $Cl(K)$. The ideal class group is an abelian group whose elements are the ideal classes of $\mathcal{O}_K$, where an ideal class is an equivalence class of fractional ideals under a specific equivalence relation. Two fractional ideals $a$ and $b$ are equivalent if there exists an element $\alpha \in K$ such that $a = \alpha b$. The group operation is multiplication of ideals.

A number field is a UFD if and only if its class number is 1. Therefore, the class number provides a quantitative measure of the failure of unique factorization. If the class number is greater than 1, it indicates that there are non-principal ideals in the ring of integers, meaning that elements cannot always be uniquely factored into irreducible elements. The calculation of class numbers is a complex and often challenging problem in algebraic number theory.