S-unit

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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[edit]

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[edit]

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[edit]

The S-unit equation is a Diophantine equation

u + v = 1

with u, v restricted to being S-units of K. The number of solutions of this equation is finite and the solutions are effectively determined using estimates for linear forms in logarithms as developed in 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 elliptic curves, and more generally superelliptic curves of the form yn=f(x).

References[edit]

Further reading[edit]