S-unit
Encyclopedia
In mathematics
, in the field of algebraic number theory
, an S
of the ring of integers
of the field. Many of the results which hold for units are also valid for S-units.
(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.
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|.
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 curve
s, and more generally so-called hyperelliptic curves of the form yn=f(x).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, an S
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
of the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
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 idealFractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed...
(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
Dirichlet's unit theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K...
holds for S-units: the group of S-units is finitely generated, with rank
Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
(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 SDiophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
- 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
Linear forms in logarithms
In number theory the method of linear forms in logarithms is the application of estimatesfor the magnitude of a finite sum\sum \beta_i\log\alpha_i=\Lambda,...
as developed in transcendence theory
Transcendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
. 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 curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s, and more generally so-called hyperelliptic curves of the form yn=f(x).