Editing Proof that e is irrational (section)

==Generalizations==
In 1840, [[Joseph Liouville|Liouville]] published a proof of the fact that ''e''<sup>2</sup> is irrational<ref>{{cite journal | last = Liouville | first = Joseph | journal = [[Journal de Mathématiques Pures et Appliquées]] | title = Sur l'irrationalité du nombre ''e'' = 2,718… | series = 1 | volume = 5 | pages = 192 | year = 1840 | language = fr}}</ref> followed by a proof that ''e''<sup>2</sup> is not a root of a second-degree polynomial with rational coefficients.<ref>{{cite journal | last = Liouville | first = Joseph | journal = [[Journal de Mathématiques Pures et Appliquées]] | title = Addition à la note sur l'irrationnalité du nombre ''e'' | series = 1 | volume = 5 | pages = 193–194 | year = 1840 | language = fr}}</ref> This last fact implies that ''e''<sup>4</sup> is irrational. His proofs are similar to Fourier's proof of the irrationality of ''e''. In 1891, [[Adolf Hurwitz|Hurwitz]] explained how it is possible to prove along the same line of ideas that ''e'' is not a root of a third-degree polynomial with rational coefficients, which implies that ''e''<sup>3</sup> is irrational.<ref>{{cite book | last1 = Hurwitz | first1 = Adolf | year = 1933 | orig-year = 1891 | title = Mathematische Werke | volume = 2 | language = de | chapter = Über die Kettenbruchentwicklung der Zahl ''e'' | publisher = [[Birkhäuser]] | location = Basel | pages = 129–133}}</ref> More generally, ''e''<sup>''q''</sup> is irrational for any non-zero rational ''q''.<ref>{{cite book | last1=Aigner | first1=Martin | author1-link = Martin Aigner | last2=Ziegler | first2=Günter M. | author2-link=Günter M. Ziegler | title=[[Proofs from THE BOOK]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1998 |pages=27–36 |isbn=978-3-642-00855-9 |doi=10.1007/978-3-642-00856-6 |edition=4th}}</ref>

[[Charles Hermite]] further proved that ''e'' is a [[transcendental number]], in 1873, which means that is not a root of any polynomial with rational coefficients, as is {{math|''e''<sup>''&alpha;''</sup>}} for any non-zero [[algebraic number|algebraic]] ''&alpha;''.<ref>{{cite journal |last=Hermite |first=C. |author-link=Charles Hermite |year=1873 |title=Sur la fonction exponentielle |lang=fr |journal=Comptes rendus de l'Académie des Sciences de Paris |volume=77 |pages=18–24}}</ref>