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The class number of a number field is by definition the order of the ideal class group of its ring of integers. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.
For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete. Infinitely many real quadratic fields with class number one Gauss conjectures that there are infinitely many real quadratic fields with class number one.
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The idea of the proof of the class number formula is most easily seen when K = Q(i).In this case, the ring of integers in K is the Gaussian integers.. An elementary manipulation shows that the residue of the Dedekind zeta function at s = 1 is the average of the coefficients of the Dirichlet series representation of the Dedekind zeta function.
In object-oriented programming, a class defines the shared aspects of objects created from the class. The capabilities of a class differ between programming languages, but generally the shared aspects consist of state and behavior that are each either associated with a particular object or with all objects of that class. [1] [2]
Google Classroom is a free blended learning platform developed by Google for educational institutions that aims to simplify creating, distributing, and grading assignments. . The primary purpose of Google Classroom is to streamline the process of sharing files between teachers and students.
Google Code Search was a free beta product from Google which debuted in Google Labs on October 5, 2006, allowing web users to search for open-source code on the Internet. Features included the ability to search using operators, namely lang: , package: , license: , and file: .
In mathematics, class number may refer to Class number (group theory), in group theory, is the number of conjugacy classes of a group; Class number (number theory), the size of the ideal class group of a number ring; Class number (binary quadratic forms), the number of equivalence classes of binary quadratic forms of a given discriminant