CLASS NUMBER Meaning and
Definition
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Class number refers to a numerical value assigned to a specific mathematical concept or structure, typically within the field of algebraic number theory. It is used to measure the properties and characteristics of certain mathematical objects known as algebraic number fields.
In algebraic number theory, an algebraic number field is an extension of the rational numbers obtained by adjoining the roots of a polynomial equation with integer coefficients. The class number of such a field is a fundamental invariant that provides information about the arithmetic behavior of the field.
More formally, the class number of an algebraic number field is defined as the number of distinct ideal classes within a given field. Here, an ideal is a generalization of the concept of a prime number, providing a way to factorize the numbers in the field. The ideal classes then represent equivalence classes of these ideals based on certain algebraic relations.
The class number captures important properties of algebraic number fields, such as the existence and uniqueness of factorizations of certain numbers in the field. It measures the extent to which unique factorization holds in the field and plays a crucial role in understanding the distribution of prime numbers within these number fields.
In summary, the class number provides a quantitative measure of the arithmetic properties of algebraic number fields, allowing mathematicians to analyze the behavior and structure of these mathematical objects.
Common Misspellings for CLASS NUMBER
- xlass number
- vlass number
- flass number
- dlass number
- ckass number
- cpass number
- coass number
- clzss number
- clsss number
- clwss number
- clqss number
- claas number
- clazs number
- claxs number
- clads number
- claes number
- claws number
- clasa number
- clasz number
- clasx number
Etymology of CLASS NUMBER
The word "class number" is derived from a combination of the term "class" and the word "number". The etymology of each term is as follows:
1. Class: The word "class" comes from the Latin word "classis", which originally referred to a division or group of citizens in ancient Rome based on their social status. Over time, the meaning of "class" has expanded to encompass various categories or divisions in different contexts, such as social classes, academic classes, or classifications in general.
2. Number: The word "number" has its roots in the Latin word "numerus", which referred to a numerical symbol or digit. It has been used to denote quantities, count, or enumerate things, and has a wide-ranging application in mathematics, statistics, and other fields.
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