{{Short description|Topic in algebraic number theory}} {{DISPLAYTITLE:''S''-unit}} {{distinguish|text=S units used to measure signal strength with an S meter}}
In mathematics, in the field of algebraic number theory, an '''''S''-unit''' generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for ''S''-units.
==Definition== Let ''K'' be a number field with ring of integers ''R''. Let ''S'' be a finite set of prime ideals of ''R''. An element ''x'' of ''K'' is an ''S''-unit if the principal fractional ideal (''x'') is a product of primes in ''S'' (to positive or negative powers). For the ring of rational integers '''Z''' one may take ''S'' to be a finite set of prime numbers and define an ''S''-unit to be a rational number whose numerator and denominator are divisible only by the primes in ''S''.
==Properties== The ''S''-units form a multiplicative group containing the units of ''R''.
Dirichlet's unit theorem holds for ''S''-units: the group of ''S''-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to ''r'' + ''s'', where ''r'' is the rank of the unit group and ''s'' = |''S''|.
==S-unit equation== The '''''S''-unit equation''' is a Diophantine equation
:''u'' + ''v'' = 1
with ''u'' and ''v'' restricted to being ''S''-units of ''K'' (or more generally, elements of a finitely generated subgroup of the multiplicative group of any field of characteristic zero). The number of solutions of this equation is finite<ref>{{Cite journal |last1=Beukers |first1=F. |last2=Schlickewei |first2=H. |date=1996 |title=The equation x+y=1 in finitely generated groups |url=http://www.impan.pl/get/doi/10.4064/aa-78-2-189-199 |journal=Acta Arithmetica |language=en |volume=78 |issue=2 |pages=189–199 |doi=10.4064/aa-78-2-189-199 |issn=0065-1036|doi-access=free }}</ref> and the solutions are effectively determined using estimates for linear forms in logarithms as developed in transcendental number theory. A variety of Diophantine equations are reducible in principle to some form of the ''S''-unit equation: a notable example is Siegel's theorem on integral points on elliptic curves, and more generally superelliptic curves of the form ''y''<sup>''n''</sup> = ''f''(''x'').
A computational solver for ''S''-unit equation is available in the software SageMath.<ref>{{Cite web|url=http://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/S_unit_solver.html|title=Solve S-unit equation x + y = 1 — Sage Reference Manual v8.7: Algebraic Numbers and Number Fields|website=doc.sagemath.org|access-date=2019-04-16}}</ref>
== References == <references /> * {{cite book | last1=Everest | first1=Graham | last2=van der Poorten | first2=Alf | author2-link=Alfred van der Poorten | last3=Shparlinski | first3=Igor | last4=Ward | first4=Thomas | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | location=Providence, RI | publisher=American Mathematical Society | year=2003 | isbn=0-8218-3387-1 | zbl=1033.11006 | pages = 19–22}} * {{cite book|first= Serge | last=Lang|authorlink = Serge Lang|title = Elliptic curves: Diophantine analysis|series = Grundlehren der mathematischen Wissenschaften|volume = 231|publisher = Springer-Verlag|year = 1978|isbn = 3-540-08489-4|pages = 128–153}} * {{cite book|first= Serge | last=Lang|authorlink = Serge Lang|title = Algebraic number theory|publisher = Springer-Verlag|isbn = 0-387-94225-4|year = 1986}} Chap. V. * {{cite book | first=Nigel | last=Smart | authorlink=N. P. Smart | title=The algorithmic resolution of Diophantine equations | series=London Mathematical Society Student Texts | volume=41 | publisher=Cambridge University Press | year=1998 | isbn=0-521-64156-X | at=[https://archive.org/details/algorithmicresol0000smar/page/ Chap. 9] | url=https://archive.org/details/algorithmicresol0000smar/page/ }} * {{cite book| first= Jürgen | last=Neukirch|authorlink = Jürgen Neukirch|title = Class field theory|series = Grundlehren der mathematischen Wissenschaften|volume = 280|publisher = Springer-Verlag|year = 1986|isbn = 3-540-15251-2|pages = 72–73}}
==Further reading== * {{cite book | first1=Alan | last1=Baker | authorlink1=Alan Baker (mathematician)| first2=Gisbert | last2= Wüstholz | authorlink2=Gisbert Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=Cambridge University Press | year=2007 | isbn=978-0-521-88268-2 }} * {{cite book | first1=Enrico | last1=Bombieri | authorlink1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=Cambridge University Press | year=2006 | isbn=978-0-521-71229-3 | zbl=1130.11034 }}
Category:Algebraic number theory