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. In algebraic number fields, the numbers are organized in a way that allows for the study of their properties and relationships. This area of mathematics is important for understanding concepts in number theory and has applications in cryptography and coding theory.