The difference between Algebraic and Transcendental

Algebraic as an adjective:

Of, or relating to, algebra.

Algebraic as an adjective (mathematics, of an [[expression]], [[equation]]{{,):

or function}} Containing only numbers, letters and arithmetic operators.

Algebraic as an adjective (algebra, number theory, of a number):

Which is a root of some polynomial whose coefficients are rational.

Algebraic as an adjective (algebra, of a field):

Whose every element is a root of some polynomial whose coefficients are rational.

Algebraic as an adjective (chess, of [[notation]]):

Describing squares by file (referred to in intrinsic order rather than by the piece starting on that file) and rank, both with reference to a fixed point rather than a player-dependent perspective.

Transcendental as a noun (obsolete):

A transcendentalist.

Transcendental as a noun (philosophy, metaphysics, Platonism, Christian theology, usually plural):

Any one of the three transcendental properties of being: truth, beauty or goodness, which respectively are the ideals of science, art and religion and the principal subjects of the study of logic, aesthetics and ethics.

Transcendental as an adjective (philosophy):

Concerned with the a priori or intuitive basis of knowledge, independent of experience.

Transcendental as an adjective:

Superior; surpassing all others; extraordinary; transcendent.

Transcendental as an adjective:

Mystical or supernatural.

Transcendental as an adjective (algebra, number theory, field theory, of a [[number]] or an [[element]] of an [[extension field]]):

Not algebraic (i.e., not the root of any polynomial that has positive degree and rational coefficients).

Transcendental as an adjective (algebra, field theory, of an [[extension field]]):

That contains elements that are not algebraic.