transcendental field extensions

A transcendental field extension occurs when you start with a base field (like the rational numbers) and expand it by adding elements that are not roots of any algebraic equation with coefficients in the base field. These new elements are transcendental, meaning they do not satisfy polynomial equations with coefficients in the original field. For example, adding the number π to the rational numbers creates a transcendental extension because π is not algebraic over the rationals. Essentially, transcendental extensions involve incorporating elements that are "beyond" algebraic solutions, resulting in larger, more complex fields.