Statement

If α and β are algebraic number

Algebraic number

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

s with α ≠ 0,1 and if β is not a rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

, then any value of α^β = exp(β log α) is a transcendental number

Transcendental number

In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

.

Comments

• The values of α and β are not restricted to real number
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
  s; complex number
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
  s are allowed (and never rational, even if both the real and imaginary parts are rational).
• In general, α^β = exp(β log α) is multivalued
  Multivalued function
  In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
  
  , where "log" stands for the complex logarithm
  Complex logarithm
  In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is log z...
  
  . This accounts for the phrase "any value of" in the theorem's statement.
• An equivalent formulation of the theorem is the following: if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then (log γ)/(log α) is either rational or transcendental. This may be expressed as saying that if log α, log γ are linearly independent
  Linear independence
  In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
  
   over the rationals, then they are linearly independent over the algebraic numbers. The generalisation of this statement to several logarithms of algebraic numbers is in the domain of transcendence theory
  Transcendence theory
  Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
  
  
• If the restriction that α and β be algebraic is removed, the statement does not remain true in general. For example Here, α is , which (as proved by the theorem itself) is transcedental rather than algebraic. Similarly, if you put α = 3 and β = (log 2)/(log 3), which is transcendental, then α^β = 2 is algebraic. A characterization of the values for α and β which yield a transcendental α^β is not known.
• Kurt Mahler
  Kurt Mahler
  Kurt Mahler was a mathematician and Fellow of the Royal Society.He was a student at the universities in Frankfurt and Göttingen, graduating with a Ph.D...
  
   proved the p-adic
  P-adic number
  In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
  
   analogue of the theorem: if α and β are in C_p, the completion of the algebraic closure
  Algebraic closure
  In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
  
   of Q_p, and they are algebraic over Q, and if |α − 1|_p < 1 and |β − 1|_p < 1, then (log_pα)/(log_pβ) is either rational or transcendental, where log_p is the p-adic logarithm function.

Corollaries

• The numbers (the Gelfond–Schneider constant) and its square root
• The number (Gelfond's constant
  Gelfond's constant
  In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by the Gelfond–Schneider theorem and noting the fact that...
  
  ), as well as

See also

• Lindemann–Weierstrass theorem
  Lindemann–Weierstrass theorem
  In mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if 1, ...,  are algebraic numbers which are linearly independent over the rational numbers ', then 1, ...,  are algebraically...
  
  
• Baker's theorem
  Baker's theorem
  In transcendence theory, a mathematical discipline, Baker's theorem gives a lower bound for linear combinations of logarithms of algebraic numbers...
  
  ; an extension of the result
• Schanuel's conjecture
  Schanuel's conjecture
  In mathematics, specifically transcendence theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers.-Statement:The conjecture is as follows:...
  
  ; if proven it would imply both the Gelfond–Schneider theorem and the Lindemann–Weierstrass theorem

External links

• A proof of the Gelfond–Schneider theorem