In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the field of
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, an ''S''-unit generalises the idea of
unit of the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
of the field. Many of the results which hold for units are also valid for ''S''-units.
Definition
Let ''K'' be a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
with ring of integers ''R''. Let ''S'' be a finite set of
prime ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
s 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 number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s and define an ''S''-unit to be a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
whose numerator and denominator are
divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
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 ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
:''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 and the solutions are effectively determined using estimates for
linear forms in logarithms as developed in
transcendental number theory
Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
Transcendenc ...
. 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 ''y''
''n'' = ''f''(''x'').
A computational solver for ''S''-unit equation is available in the software
SageMath.
References
*
*
* Chap. V.
*
*
Further reading
*
* {{cite book , first1=Enrico , last1=Bombieri , authorlink1=Enrico Bombieri , first2=Walter , last2=Gubler , title=Heights in Diophantine Geometry , series=New Mathematical Monographs , volume=4 , publisher=
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, year=2006 , isbn=978-0-521-71229-3 , zbl=1130.11034
Algebraic number theory