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{{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 }} 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 yⁿ = f(x).

A computational solver for S-unit equation is available in the software SageMath.{{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}}

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}}

S-unit

Definition

Properties

S-unit equation

References

Further reading