S-unit

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 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|.

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 is Siegel's theorem on integral points on curves.

References

• Graham Everest; Alf van der Poorten, Igor Shparlinksi, Thomas Ward (2003). Recurrence sequences. American Mathematical Society, 19-22. ISBN 0-8218-3387-1. 
• Serge Lang (1978). Elliptic curves: Diophantine analysis, 128-153. ISBN 3-540-08489-4. 
• Serge Lang (1986). Algebraic number theory. Springer. ISBN 0-387-94225-4.  Chap. V.
• N.P. Smart (1998). The algorithmic resolution of Diophantine equations. Cambridge University Press. ISBN 0-521-64156-X.  Chap. 9.
• Jürgen Neukirch (1986). Class field theory, 72-73. ISBN 3-540-12521-2.