fundamental units

The ring R of algebraic integersMathworldPlanetmath of any algebraic number fieldMathworldPlanetmath contains a finite setMathworldPlanetmath H={η1,η2,,ηt} of so-called fundamental unitsMathworldPlanetmath such that every unit ε of R is a power ( productPlanetmathPlanetmath of these, multiplied by a root of unityMathworldPlanetmath:


Conversely, every such element ε of the field is a unit of R.

Examples:  units of quadratic fields,  units of certain cubic fields (

For some algebraic number fields, such as all imaginary quadratic fieldsMathworldPlanetmath, the set H may be empty (t=0).  In the case of a single fundamental unit (t=1), which occurs e.g. in all real quadratic fields (, there are two alternative units η and its conjugatePlanetmathPlanetmath η¯ which one can use as fundamental unit; then we can speak of the uniquely determined fundamental unit η1 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