In mathematics, a cyclotomic unit (or circular unit) is a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
of an
algebraic 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 f ...
which is the product of numbers of the form (ζ − 1) for ζ an ''n''
th root of unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
and 0 < ''a'' < ''n''.
Properties
The cyclotomic units form a subgroup of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
in the
group of units
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the element is unique for this ...
of a
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
. The index of this subgroup of ''real'' cyclotomic units (those cyclotomic units in the maximal real subfield) within the full real unit group is equal to the
class number of the maximal real subfield of the
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
.
* If is the power of a prime, then is not a unit; however the numbers for , and ±ζ generate the group of cyclotomic units.
* If is a
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
having two or more distinct prime factors, then is a unit. The subgroup of cyclotomic units generated by with is not of finite index in general.
The cyclotomic units satisfy ''distribution relations''. Let be a rational number prime to and let denote . Then for we have
Using these distribution relations and the symmetry relation a basis ''B''
''n'' of the cyclotomic units can be constructed with the property that for .
See also
*
Elliptic unit
*
Modular unit
Notes
References
*
*
*
Algebraic number theory
Cyclotomic fields
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