A field extension is a concept in abstract algebra where a larger field is created from a smaller field by adding new elements. This process allows for the solution of equations that cannot be solved within the original field. For example, the field of rational numbers, Q, can be extended to include the square root of 2, forming the field Q(√2). Field extensions are important in various areas of mathematics, including number theory and algebraic geometry. They help mathematicians understand the properties of numbers and functions by providing a broader context in which to work. Extensions can be finite or infinite, depending on the number of new elements added.