A field extension is a way to create a new field from an existing one by adding new elements that satisfy certain algebraic properties. In mathematics, a field is a set equipped with two operations, addition and multiplication, that follow specific rules. When we extend a field, we include new elements that allow us to solve equations that were previously unsolvable within the original field. For example, the field of rational numbers, denoted as ℚ, can be extended to include the square root of 2, forming the field ℚ(√2). This new field contains all numbers that can be expressed as a combination of rational numbers and √2, allowing for more solutions to polynomial equations.