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 |
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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 |