An algebraic extension is a type of field extension in mathematics where a new field is created by adding roots of polynomial equations from a base field. For example, if you start with the field of rational numbers, adding the square root of 2 creates an algebraic extension. This new field contains all numbers that can be expressed as combinations of rational numbers and the square root of 2. In algebra, an extension is considered algebraic if every element in the new field is a solution to some polynomial equation with coefficients from the original field. This concept is essential in areas like Galois theory and helps in understanding the solvability of polynomial equations.