A number field is a mathematical structure that extends the concept of rational numbers. It consists of a set of numbers that includes rational numbers and additional elements, allowing for operations like addition, subtraction, multiplication, and division. Number fields are often used in algebra and number theory to study properties of numbers and their relationships. One common example of a number field is the field of rational numbers combined with the square root of a prime number, such as √2. This creates a new set of numbers, including expressions like a + b√2, where a and b are rational numbers. Number fields help mathematicians understand more complex number systems and their applications.