A field extension is a concept in abstract algebra, specifically in the study of field theory. It refers to a larger field that contains a smaller field as a subset. This allows mathematicians to explore more complex structures and relationships between numbers. For example, the field of rational numbers, Q, can be extended to include the square root of 2, forming the field of real numbers, R. Field extensions can be classified into different types, such as finite extensions and infinite extensions. They are essential for solving polynomial equations and understanding the properties of numbers. By studying field extensions, mathematicians can gain insights into the behavior of algebraic structures and their applications in various areas of mathematics.