An algebraic number field is a special type of mathematical structure that extends the concept of rational numbers. It is formed by taking a set of numbers that can be expressed as solutions to polynomial equations with rational coefficients. These fields include all rational numbers and can contain irrational numbers, such as square roots or cube roots, that satisfy specific polynomial equations. The study of algebraic number fields is a key area in number theory, which explores the properties and relationships of numbers. Important concepts related to these fields include algebraic integers, which are the roots of monic polynomials, and the ring of integers within the field, which consists of all algebraic integers in that field.