Editing Gelfond–Schneider theorem (section)

===Comments===

The values of ''a'' and ''b'' are not restricted to [[real number]]s; [[complex number]]s are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).

In general, {{nowrap|1=''a<sup>b</sup>'' = exp(''b'' log ''a'')}} is [[multivalued function|multivalued]], where log stands for the complex [[natural logarithm]]. (This is the multivalued inverse of the exponential function exp.) 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 {{nowrap|(log ''γ'')/(log ''α'')}} is either rational or transcendental.  This may be expressed as saying that if {{nowrap|log ''α''}}, {{nowrap|log ''γ''}} are [[linear independence|linearly independent]] over the rationals, then they are linearly independent over the algebraic numbers.  The generalisation of this statement to more general [[linear forms in logarithms]] of several algebraic numbers is in the domain of [[transcendental number theory]].

If the restriction that ''a'' and ''b'' be algebraic is removed, the statement does not remain true in general. For example,
<math display="block">{\left(\sqrt{2}^{\sqrt{2}}\right)}^{\sqrt{2}} = \sqrt{2}^{\sqrt{2} \cdot \sqrt{2}} = \sqrt{2}^2 = 2.</math>
Here, ''a'' is {{radic|2}}<sup>{{radic|2}}</sup>, which (as proven by the theorem itself) is transcendental rather than algebraic. Similarly, if {{nowrap|1=''a'' = 3}} and {{nowrap|1=''b'' = (log 2)/(log 3)}}, which is transcendental, then {{nowrap|1=''a<sup>b</sup>'' = 2}} is algebraic. A characterization of the values for ''a'' and ''b'' which yield a transcendental ''a<sup>b</sup>'' is not known.

[[Kurt Mahler]] proved the [[P-adic number|''p''-adic]] analogue of the theorem: if ''a'' and ''b'' are in '''C'''<sub>''p''</sub>, the [[Complete metric space#Completion|completion]] of the [[algebraic closure]] of '''Q'''<sub>''p''</sub>, and they are algebraic over '''Q''', and if <math>|a-1|_p<1</math> and <math>|b-1|_p<1,</math> then <math>(\log_p a)/(\log_p b)</math> is either rational or transcendental, where log<sub>''p''</sub> is the [[P-adic exponential function#p-adic logarithm function|''p''-adic logarithm function]].