Transcendence theory

Transcendence theory studies numbers that are not roots of nonzero polynomials with rational coefficients. It proves such numbers transcendental and investigates their algebraic independence.

Algebraic number: A complex number that is a root of a nonzero polynomial with rational coefficients. Transcendence is defined by exclusion from this class.
Liouville's theorem (transcendence theory): A theorem bounding how closely an algebraic irrational can be approximated by rational numbers. It gives a criterion that proves certain explicitly constructed numbers transcendental.
Liouville number: A real number with rational approximations arbitrarily more accurate than any fixed power of the denominator. Liouville's theorem proves these numbers transcendental.
Joseph Liouville: A French mathematician who proved the first explicit existence results for transcendental numbers. His approximation theorem initiated systematic constructions and proofs of transcendence.
Algebraic independence of π and e: The unresolved question of whether π and e satisfy any nonzero polynomial relation over the rationals. Both constants are transcendental, but their joint algebraic independence is unknown.
Transcendental number: A number that is not algebraic over the rational numbers. These are the objects the theory aims to identify and compare.
Roth's theorem: A theorem stating that algebraic irrational numbers admit only finitely many rational approximations beyond a specified quality. Its sharp approximation bound exemplifies how arithmetic constraints distinguish algebraic numbers.
e (mathematical constant): The base of the natural logarithm, approximately 2.71828. Its transcendence was established by Hermite using analytic estimates.
Charles Hermite: A French mathematician whose work established the transcendence of e. His 1873 proof introduced techniques that shaped later transcendence arguments.
Euler's constant: The constant γ defined as the limiting difference between the harmonic sums and the natural logarithm. Its irrationality and transcendence remain unproved.

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Tijdeman's theoremNarrower topic: The theorem imports transcendence methods into a problem about integer powers.