NUMBER FIELD Meaning and
Definition
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A number field is a fundamental concept in the field of algebraic number theory. It refers to a finite extension of the rational numbers, typically denoted as Q, which is the field of rational numbers. In other words, a number field is an extension field of the rational numbers where additional elements, called algebraic numbers, are adjoined in order to provide a larger set of numbers.
A number field can be thought of as an algebraic structure that encompasses both the rational numbers and the new elements introduced through the extension. These new elements are typically solutions to polynomial equations with rational coefficients. Therefore, a number field contains all the roots of these polynomials, making it a rich and encompassing field.
The elements of a number field possess specific properties such as addition, subtraction, multiplication, and division, allowing for the execution of arithmetic operations within the field. They also possess distinct algebraic properties, such as conjugation and norms, that relate to the structure and behavior of the numbers in the field.
Number fields have numerous significant applications in various branches of mathematics, particularly in algebraic geometry, algebraic topology, and cryptography. They are also crucial in studying Diophantine equations, the distribution of prime numbers, and the structure of number rings. Overall, number fields provide a key framework for exploring and understanding the properties and behavior of numbers beyond the rational realm.
Common Misspellings for NUMBER FIELD
- bumber field
- mumber field
- jumber field
- humber field
- nymber field
- nhmber field
- njmber field
- nimber field
- n8mber field
- n7mber field
- nunber field
- nukber field
- nujber field
- numver field
- numner field
- numher field
- numger field
- numbwr field
- numbsr field
- numbdr field
Etymology of NUMBER FIELD
The word "number field" is derived from two components: "number" and "field".
The term "number" originates from the Latin word "numerus", which means "number". It has its roots in Indo-European and is related to the Sanskrit word "namati", meaning "he assigns". Over time, "number" came to refer to the concept of a numerical value or quantity.
The term "field" comes from the Old English word "feld", meaning "open land" or "plain". It shares the same Germanic root as the Dutch word "veld" and the German word "Feld". In mathematics, a "field" describes a set of elements with operations like addition, subtraction, multiplication, and division—similar to the arithmetical operations on numbers.
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