Editing Roth's theorem (section)

== Generalizations ==

There is a higher-dimensional version, [[subspace theorem|Schmidt's subspace theorem]], of the basic result. There are also numerous extensions, for example using the [[p-adic metric|''p''-adic metric]],<ref>{{citation | first=D. | last=Ridout | title=The ''p''-adic generalization of the Thue–Siegel–Roth theorem | journal=[[Mathematika]] | volume=5 | pages=40–48 | year=1958 | zbl=0085.03501 | doi=10.1112/s0025579300001339}}</ref> based on the Roth method.

[[William J. LeVeque]] generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed [[algebraic number field]].  Define the ''[[height function|height]]'' ''H''(ξ) of an algebraic number ξ to be the maximum of the [[absolute value]]s of the [[coefficient]]s of its [[minimal polynomial (field theory)|minimal polynomial]].  Fix κ>2.  For a given algebraic number α and algebraic number field ''K'', the equation

:<math>|\alpha - \xi| < \frac{1}{H(\xi)^\kappa}</math>

has only finitely many solutions in elements ξ of ''K''.<ref>{{citation | last = LeVeque | first = William J. | author-link = William J. LeVeque | title = Topics in Number Theory, Volumes I and II | publisher = Dover Publications | location = New York | year = 2002 | orig-year = 1956 | isbn = 978-0-486-42539-9 | zbl = 1009.11001 | pages = [https://archive.org/details/topicsinnumberth0000leve/page/ II:148–152] | url = https://archive.org/details/topicsinnumberth0000leve/page/ }}</ref>