A transcendental extension is a type of field extension in mathematics where new elements are added to a field that are not roots of any polynomial with coefficients from the original field. This means that the new elements cannot be expressed as solutions to algebraic equations involving the original field's elements. For example, if we start with the field of rational numbers, adding the number π creates a transcendental extension because π is not a solution to any polynomial equation with rational coefficients. Such extensions are important in fields like algebra and number theory, as they help in understanding the structure of different number systems.