fundamental units
The ring of algebraic integers of any algebraic number
field
contains a finite set
of so-called
fundamental units
such that every unit of
is a power (http://planetmath.org/GeneralAssociativity) product
of
these, multiplied by a root of unity
:
Conversely, every such element of the field is a unit of .
Examples: units of quadratic fields, units of certain cubic fields (http://planetmath.org/UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding)
For some algebraic number fields, such as all imaginary
quadratic fields, the set may be empty (). In the
case of a single fundamental unit (), which occurs e.g.
in all
real quadratic fields (http://planetmath.org/ImaginaryQuadraticField),
there are two alternative units
and its conjugate
which one can use as
fundamental unit; then we can speak of the uniquely
determined fundamental unit which is greater than 1.
Title | fundamental units |
---|---|
Canonical name | FundamentalUnits |
Date of creation | 2014-11-24 16:38:36 |
Last modified on | 2014-11-24 16:38:36 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 22 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 11R27 |
Classification | msc 11R04 |
Related topic | NumberField |
Related topic | AlgebraicInteger |