- S-unit
In
mathematics , in the field ofalgebraic 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 "R" is an "S"-unit if the prime ideals dividing ("x") are all in "S". For the ring of rational integers Z one may also take "S" to be a finite set of prime numbers and define an "S"-unit to be an integer 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"|.-unit equation
The "S"
-unit equation is a Diophantine equation :"u" + "v" = 1
with "u", "v" restricted to being "S"-units of "R". The number of solutions of this equation is finite and the solutions are effectively determined using
transcendence theory . A variety of Diophantine equations are reducible in principle to some form of the "S"-unit equation: a notable example isSiegel's theorem on integral points on curves.References
*cite book | author=Graham Everest | coauthors=Alf van der Poorten, Igor Shparlinksi, Thomas Ward | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | publisher=
American Mathematical Society | year=2003 | isbn=0-8218-3387-1 | pages=19-22
*
*cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=Springer | isbn=0-387-94225-4 | year=1986 Chap. V.
* Chap. 9.
*
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