An algebraic number field is a special type of mathematical structure that extends the concept of rational numbers. It is formed by taking a rational number and adding solutions to polynomial equations with integer coefficients. This means that algebraic number fields contain not only rational numbers but also numbers that can be expressed as roots of these polynomials. For example, the field of rational numbers, denoted as Q, can be extended to include numbers like the square root of 2, which is a solution to the polynomial equation x^2 - 2 = 0. Such extensions create new fields, like Q(√2), which are essential in various areas of mathematics, including number theory and algebra.