Ring Of Integers
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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 ...
, 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 In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
of a
monic polynomial In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
with
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s: x^n+c_x^+\cdots+c_0. This ring is often denoted by O_K or \mathcal O_K. Since any
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
belongs to K and is an integral element of K, the ring \mathbb is always a subring of O_K. The ring of integers \mathbb is the simplest possible ring of integers. Namely, \mathbb=O_ where \mathbb is the field of
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 ...
s. And indeed, in
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 ...
the elements of \mathbb are often called the "rational integers" because of this. The next simplest example is the ring of
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf ...
s \mathbb /math>, consisting of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s whose real and imaginary parts are integers. It is the ring of integers in the number field \mathbb(i) of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, \mathbb /math> is a Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.


Properties

The ring of integers is a finitely-generated - module. Indeed, it is a free -module, and thus has an integral basis, that is a basis of the -
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
  such that each element  in can be uniquely represented as :x=\sum_^na_ib_i, with .Cassels (1986) p. 193 The rank  of as a free -module is equal to the degree of  over .


Examples


Computational tool

A useful tool for computing the integral closure of the ring of integers in an algebraic field is the discriminant. If is of degree over , and \alpha_1,\ldots,\alpha_n \in \mathcal_K form a basis of over , set d = \Delta_(\alpha_1,\ldots,\alpha_n). Then, \mathcal_K is a submodule of the spanned by \alpha_1/d,\ldots,\alpha_n/d. pg. 33 In fact, if is square-free, then \alpha_1,\ldots,\alpha_n forms an integral basis for \mathcal_K. pg. 35


Cyclotomic extensions

If is a prime,  is a th
root of unity In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...
and is the corresponding cyclotomic field, then an integral basis of \mathcal_K=\bf zeta/math> is given by .


Quadratic extensions

If d is a square-free integer and K = \mathbb(\sqrt\,) is the corresponding
quadratic field In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...
, then \mathcal_K is a ring of quadratic integers and its integral basis is given by (1, \frac) if and by (1, \sqrt) if . This can be found by computing the minimal polynomial of an arbitrary element a + b\sqrt \in \mathbf(\sqrt) where a,b \in \mathbf.


Multiplicative structure

In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers , the element 6 has two essentially different factorizations into irreducibles: : 6 = 2 \cdot 3 = (1 + \sqrt)(1 - \sqrt). A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals. The units of a ring of integers is a finitely generated abelian group by Dirichlet's unit theorem. The
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group ...
consists of the roots of unity of . A set of torsion-free generators is called a set of '' fundamental units''.


Generalization

One defines the ring of integers of a non-archimedean local field as the set of all elements of with absolute value ; this is a ring because of the strong triangle inequality. If is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion. For example, the -adic integers are the ring of integers of the -adic numbers .


See also

* Minimal polynomial (field theory) * Integral closure – gives a technique for computing integral closures


Notes


Citations


References

* * * * {{refend Ring theory Algebraic number theory