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

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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

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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

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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[1] 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.[2]

References

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  1. ^ Beukers, F.; Schlickewei, H. (1996). "The equation x+y=1 in finitely generated groups". Acta Arithmetica. 78 (2): 189–199. doi:10.4064/aa-78-2-189-199. ISSN 0065-1036.
  2. ^ "Solve S-unit equation x + y = 1 — Sage Reference Manual v8.7: Algebraic Numbers and Number Fields". doc.sagemath.org. Retrieved 2019-04-16.

Further reading

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