Editing Transcendental number theory (section)

==History==
===Approximation by rational numbers: Liouville to Roth===

Use of the term ''transcendental'' to refer to an object that is not algebraic dates back to the seventeenth century, when [[Gottfried Leibniz]] proved that the [[sine function]] was not an [[algebraic function]].<ref>N. Bourbaki, ''Elements of the History of Mathematics'' Springer (1994).</ref>  The question of whether certain classes of numbers could be transcendental dates back to 1748<ref>{{Harvnb|Gelfond|1960|p=2}}.</ref> when [[Euler]] asserted<ref>{{cite book |first=L. |last=Euler |title=Introductio in analysin infinitorum |url=https://archive.org/details/bub_gb_jQ1bAAAAQAAJ |location=Lausanne |year=1748 }}</ref> that the number log<sub>''a''</sub>''b'' was not [[algebraic number|algebraic]] for [[rational number]]s ''a'' and ''b'' provided ''b'' is not of the form ''b''&nbsp;=&nbsp;''a''<sup>''c''</sup> for some rational ''c''.

Euler's assertion was not proved until the twentieth century, but almost a hundred years after his claim [[Joseph Liouville]] did manage to prove the existence of numbers that are not algebraic, something that until then had not been known for sure.<ref>The existence proof based on the different [[cardinalities]] of the [[real number|real]] and the [[algebraic number|algebraic]] numbers was not possible before [[Cantor's first set theory article]] in 1874.</ref>  His original papers on the matter in the 1840s sketched out arguments using [[simple continued fraction]]s to construct transcendental numbers.  Later, in the 1850s, he gave a [[Necessary and sufficient condition|necessary condition]] for a number to be algebraic, and thus a sufficient condition for a number to be transcendental.<ref>{{cite journal | last1 = Liouville | first1 = J. | year = 1844 | title = Sur les classes très étendues de quantités dont la valeur n'est ni algébrique ni même réductible à des irrationelles algébriques | journal = Comptes rendus de l'Académie des Sciences de Paris | volume = 18 | pages = [http://gallica.bnf.fr/ark:/12148/bpt6k2977n/f883.image 883–885], [http://gallica.bnf.fr/ark:/12148/bpt6k2977n/f910.image 910–911]}}; ''Journal Math. Pures et Appl.'' '''16''', (1851), pp.133–142.</ref>  This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the number [[Euler's number|''e'']] is transcendental.  But his work did provide a larger class of transcendental numbers, now known as [[Liouville number]]s in his honour.

Liouville's criterion essentially said that algebraic numbers cannot be very well approximated by rational numbers.  So if a number can be very well approximated by rational numbers then it must be transcendental.  The exact meaning of "very well approximated" in Liouville's work relates to a certain exponent.  He showed that if α is an [[algebraic number]] of degree ''d''&nbsp;≥&nbsp;2 and ε is any number greater than zero, then the expression
:<math>\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{d+\varepsilon}}</math>
can be satisfied by only finitely many rational numbers ''p''/''q''.  Using this as a criterion for transcendence is not trivial, as one must check whether there are infinitely many solutions ''p''/''q'' for every ''d''&nbsp;≥&nbsp;2.

In the twentieth century work by [[Axel Thue]],<ref>{{cite journal |first=A. |last=Thue |title=Über Annäherungswerte algebraischer Zahlen |journal=[[Journal für die reine und angewandte Mathematik|J. Reine Angew. Math.]] |volume=1909 |year=1909 |issue= 135|pages=284–305 |doi=10.1515/crll.1909.135.284 |s2cid=125903243 }}</ref> [[Carl Ludwig Siegel|Carl Siegel]],<ref>{{cite journal |first=C. L. |last=Siegel |title=Approximation algebraischer Zahlen |journal=[[Mathematische Zeitschrift]] |volume=10 |year=1921 |issue=3–4 |pages=172–213 |doi=10.1007/BF01211608 | doi-access=free |url=https://zenodo.org/record/1538156 }}</ref> and [[Klaus Roth]]<ref>{{cite journal |first=K. F. |last=Roth |title=Rational approximations to algebraic numbers |journal=[[Mathematika]] |volume=2 |year=1955 |issue=1 |pages=1–20 |doi=10.1112/S0025579300000644 }} And "Corrigendum", p. 168, {{doi|10.1112/S002559300000826}}.</ref> reduced the exponent in Liouville's work from ''d''&nbsp;+&nbsp;ε to ''d''/2&nbsp;+&nbsp;1&nbsp;+&nbsp;ε, and finally, in 1955, to 2&nbsp;+&nbsp;ε.  This result, known as the [[Thue–Siegel–Roth theorem]], is ostensibly the best possible, since if the exponent 2&nbsp;+&nbsp;ε is replaced by just 2 then the result is no longer true.  However, [[Serge Lang]] conjectured an improvement of Roth's result; in particular he conjectured that ''q''<sup>2+ε</sup> in the denominator of the right-hand side could be reduced to <math>q^{2}(\log q)^{1+ \epsilon}</math>.

Roth's work effectively ended the work started by Liouville, and his theorem allowed mathematicians to prove the transcendence of many more numbers, such as the [[Champernowne constant]].  The theorem is still not strong enough to detect ''all'' transcendental numbers, though, and many famous constants including ''e'' and π either are not or are not known to be very well approximable in the above sense.<ref>{{cite journal |first=K. |last=Mahler |title=On the approximation of π |journal=Proc. Akad. Wetensch. Ser. A |volume=56 |year=1953 |pages=30–42 }}</ref>

===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>

===Other techniques: Cantor and Zilber===

In the 1870s, [[Georg Cantor]] started to develop [[set theory]] and, in 1874, published a [[Georg Cantor's first set theory article|paper]] proving that the algebraic numbers could be put in [[Bijection|one-to-one correspondence]] with the set of [[natural number]]s, and thus that the set of transcendental numbers must be [[uncountable set|uncountable]].<ref>{{cite journal |first=G. |last=Cantor |url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=266194 |title=Ueber eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen |journal=[[Journal für die reine und angewandte Mathematik|J. Reine Angew. Math.]] |volume=1874 |year=1874 |issue= 77|pages=258–262 |doi=10.1515/crll.1874.77.258 |s2cid=199545885 |language=de}}</ref>  Later, in 1891, Cantor used his more familiar [[Cantor's diagonal argument|diagonal argument]] to prove the same result.<ref>{{cite journal |first=G. |last=Cantor |title=Ueber eine elementare Frage der Mannigfaltigkeitslehre |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=1 |year=1891 |pages=75–78 |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002113910 |language=de}}</ref>  While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number,<ref>{{cite book |first1=M. |last1=Kac |first2=U. |last2=Stanislaw |title=Mathematics and Logic |url=https://archive.org/details/mathematicslogic0000kacm |url-access=registration |publisher=Fredering A. Praeger |year=1968 |page=[https://archive.org/details/mathematicslogic0000kacm/page/13 13] }}</ref><ref>{{cite book |first=E. T. |last=Bell |title=Men of Mathematics |location=New York |publisher=Simon & Schuster |year=1937 |page=[https://archive.org/details/menofmathematics0041bell/page/569 569] |title-link=Men of Mathematics }}</ref> the proofs in both the aforementioned papers give methods to construct transcendental numbers.<ref>{{cite journal |first=R. |last=Gray |title=Georg Cantor and Transcendental Numbers |url = http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf |journal=[[American Mathematical Monthly]] |volume=101 |year=1994 |issue=9 |pages=819–832 |jstor=2975129 |doi=10.1080/00029890.1994.11997035 }}</ref>

While Cantor used set theory to prove the plenitude of transcendental numbers, a recent development has been the use of [[model theory]] in attempts to prove an [[unsolved problem]] in transcendental number theory.  The problem is to determine the [[transcendence degree]] of the field
:<math>K=\mathbb{Q}(x_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n})</math>
for complex numbers ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> that are linearly independent over the rational numbers.  [[Stephen Schanuel]] [[Schanuel's conjecture|conjectured]] that the answer is at least ''n'', but no proof is known.  In 2004, though, [[Boris Zilber]] published a paper that used model theoretic techniques to create a structure that behaves very much like the [[complex number]]s equipped with the operations of addition, multiplication, and exponentiation.  Moreover, in this abstract structure Schanuel's conjecture does indeed hold.<ref>{{cite journal |first=B. |last=Zilber |title=Pseudo-exponentiation on algebraically closed fields of characteristic zero |journal=Annals of Pure and Applied Logic |volume=132 |year=2005 |issue=1 |pages=67–95 |mr=2102856 |doi=10.1016/j.apal.2004.07.001 |doi-access=free }}</ref>  Unfortunately it is not yet known that this structure is in fact the same as the complex numbers with the operations mentioned; there could exist some other abstract structure that behaves very similarly to the complex numbers but where Schanuel's conjecture doesn't hold.  Zilber did provide several criteria that would prove the structure in question was '''C''', but could not prove the so-called Strong Exponential Closure axiom.  The simplest case of this axiom has since been proved,<ref>{{cite journal |first=D. |last=Marker |title=A remark on Zilber's pseudoexponentiation |journal=Journal of Symbolic Logic |volume=71 |issue=3 |year=2006 |pages=791–798 |mr=2250821 |jstor=27588482 |doi=10.2178/jsl/1154698577|s2cid=1477361 }}</ref> but a proof that it holds in full generality is required to complete the proof of the conjecture.