ALGEBRAIC INTEGER Meaning and
Definition
-
An algebraic integer refers to a complex number that is a root of a monic polynomial equation with integer coefficients. In other words, it is a solution to an algebraic equation where the coefficients are integers, and the leading coefficient is equal to 1. More formally, let ?_? be an algebraic integer if there exists a monic polynomial ?(?) with integer coefficients such that ?(?_?) = 0.
Algebraic integers are a fundamental concept in number theory and algebraic number theory, serving as a generalization of the notion of an integer to a larger set of numbers. They lie in algebraic number fields and can be considered as the analogs of integers in these fields.
A key property of algebraic integers is that they are closed under addition, subtraction, and multiplication. This means that if two algebraic integers are added, subtracted, or multiplied together, the resulting number remains an algebraic integer. However, this closure property does not hold for division, as the quotient of two algebraic integers may not be an algebraic integer.
Furthermore, algebraic integers form a ring, known as the ring of algebraic integers, which is a commutative ring with unity. This ring has various algebraic properties, such as being an integral domain and a unique factorization domain.
Etymology of ALGEBRAIC INTEGER
The word "algebraic integer" has its origins in the field of algebraic number theory, which deals with properties and structures of numbers that are roots of polynomial equations with integer coefficients.
The term "algebraic" is derived from the word "algebra", which, in its earliest usage, referred to calculations and operations involving variables and unknown quantities. The word "integer" comes from the Latin word "integer", meaning "whole" or "untouched". In mathematics, an integer refers to a positive or negative whole number, including zero.
The combination of the terms "algebraic" and "integer" is used to describe a number that is a root of a polynomial equation with integer coefficients, meaning it satisfies a polynomial equation where all the coefficients are integers. These algebraic integers are an important concept in algebraic number theory as they generalize the concept of whole numbers to more complex number systems.
