Number fields are extensions of the rational numbers, characterized by the addition of roots of polynomials with rational coefficients. They serve as a fundamental structure in algebraic number theory, enabling the study of number properties through algebraic methods.

Number Fields

Number fields

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. They include integers, finite decimals, and repeating decimals, providing a comprehensive understanding of how numbers can be represented in fractional form.

rational numbers

Roots of polynomials are the values of the variable that satisfy the polynomial equation, making the polynomial equal to zero. These roots can be real or complex numbers, and their properties are central to understanding the behavior and characteristics of polynomial functions.

roots of polynomials

Rational coefficients are numbers that can be expressed as the quotient of two integers, where the denominator is not zero, and are used in algebraic expressions and equations to allow for precise manipulation and solution. They are essential in polynomial equations, enabling operations such as addition, subtraction, multiplication, and division to be performed while maintaining the rationality of the coefficients.

rational coefficients

Algebraic number theory is a branch of number theory that studies the properties of algebraic numbers, which are roots of non-zero polynomial equations with rational coefficients. It connects number theory with abstract algebra, particularly through the use of field theory, Galois theory, and ring theory to solve problems related to integers and their generalizations.

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