transcendental extensions

A transcendental extension is a concept in mathematics, specifically in field theory, which deals with expanding a set of numbers. When we add new elements to a field (like rational numbers) that are not solutions to any polynomial equation with coefficients from that field, we create a transcendental extension. For example, the number π (pi) is transcendental because it cannot be expressed as a root of any polynomial with rational coefficients. This contrasts with algebraic extensions, where the new elements are roots of such polynomials. Transcendental extensions help in understanding the complexity of numbers beyond algebraic relationships.