Editing Transcendental number theory (section)

===Auxiliary functions: Hermite to Baker===
Fortunately other methods were pioneered in the nineteenth century to deal with the algebraic properties of ''e'', and consequently of π through [[Euler's identity]].  This work centred on use of the so-called [[auxiliary function]].  These are [[Function (mathematics)|functions]] which typically have many zeros at the points under consideration.  Here "many zeros" may mean many distinct zeros, or as few as one zero but with a high [[Multiplicity (mathematics)#Multiplicity of a zero of a function|multiplicity]], or even many zeros all with high multiplicity.  [[Charles Hermite]] used auxiliary functions that approximated the functions <math>e^{kx}</math> for each [[natural number]] <math>k</math> in order to prove the transcendence of <math>e</math> in 1873.<ref>{{cite journal |first=C. |last=Hermite |title=Sur la fonction exponentielle |journal=C. R. Acad. Sci. Paris |volume=77 |year=1873 }}</ref>  His work was built upon by [[Ferdinand von Lindemann]] in the 1880s<ref>{{cite journal |first=F. |last=Lindemann |title=Ueber die Zahl π  |journal=[[Mathematische Annalen]] |volume=20 |year=1882 |issue=2 |pages=213–225 |doi=10.1007/BF01446522 | doi-access=free |url=https://zenodo.org/record/1428234 }}</ref> in order to prove that ''e''<sup>α</sup> is transcendental for nonzero algebraic numbers α.  In particular this proved that π is transcendental since ''e''<sup>π''i''</sup> is algebraic, and thus answered in the negative the [[Compass and straightedge constructions|problem of antiquity]] as to whether it was possible to [[Squaring the circle|square the circle]].  [[Karl Weierstrass]] developed their work yet further and eventually proved the [[Lindemann–Weierstrass theorem]] in 1885.<ref>{{cite journal |first=K. |last=Weierstrass |title=Zu Hrn. Lindemann's Abhandlung: 'Über die Ludolph'sche Zahl' |journal=Sitzungber. Königl. Preuss. Akad. Wissensch. Zu Berlin |volume=2 |year=1885 |pages=1067–1086 }}</ref>

In 1900 [[David Hilbert]] posed his famous [[Hilbert's problems|collection of problems]].  The [[Hilbert's seventh problem|seventh of these]], and one of the hardest in Hilbert's estimation, asked about the transcendence of numbers of the form ''a''<sup>''b''</sup> where ''a'' and ''b'' are algebraic, ''a'' is not zero or one, and ''b'' is [[irrational number|irrational]].  In the 1930s [[Alexander Gelfond]]<ref>{{cite journal |first=A. O. |last=Gelfond |title=Sur le septième Problème de D. Hilbert |journal=Izv. Akad. Nauk SSSR |volume=7 |year=1934 |pages=623–630 }}</ref> and [[Theodor Schneider]]<ref>{{cite journal |first=T. |last=Schneider |title=Transzendenzuntersuchungen periodischer Funktionen. I. Transzendend von Potenzen |journal=[[Journal für die reine und angewandte Mathematik]] |volume=1935 |year=1935 |issue= 172|pages=65–69 |doi=10.1515/crll.1935.172.65 |s2cid=115310510 }}</ref> proved that all such numbers were indeed transcendental using a non-explicit auxiliary function whose existence was granted by [[Siegel's lemma]].  This result, the [[Gelfond–Schneider theorem]], proved the transcendence of numbers such as [[Gelfond's constant|''e''<sup>π</sup>]] and the [[Gelfond–Schneider constant]].

The next big result in this field occurred in the 1960s, when [[Alan Baker (mathematician)|Alan Baker]] made progress on a problem posed by Gelfond on [[linear forms in logarithms]].  Gelfond himself had managed to find a non-trivial lower bound for the quantity
:<math>|\beta_1\log\alpha_1 +\beta_2\log\alpha_2|\,</math>
where all four unknowns are algebraic, the αs being neither zero nor one and the βs being irrational.  Finding similar lower bounds for the sum of three or more logarithms had eluded Gelfond, though.  The proof of [[Baker's theorem]] contained such bounds, solving Gauss' [[class number problem]] for class number one in the process.  This work won Baker the [[Fields medal]] for its uses in solving [[Diophantine equation]]s.  From a purely transcendental number theoretic viewpoint, Baker had proved that if α<sub>1</sub>, ..., α<sub>''n''</sub> are algebraic numbers, none of them zero or one, and β<sub>1</sub>, ..., β<sub>''n''</sub> are algebraic numbers such that 1, β<sub>1</sub>, ..., β<sub>''n''</sub> are [[linearly independent]] over the rational numbers, then the number
:<math>\alpha_1^{\beta_1}\alpha_2^{\beta_2}\cdots\alpha_n^{\beta_n}</math>
is transcendental.<ref>A. Baker, ''Linear forms in the logarithms of algebraic numbers. I, II, III'', Mathematika '''13''' ,(1966), pp.204–216; ibid. '''14''', (1967), pp.102–107; ibid. '''14''', (1967), pp.220–228, {{MathSciNet | id = 0220680}}</ref>