fundamental units

The ring $R$ of algebraic integers of any algebraic number field contains a finite set $H=\{{\eta }_1,{\eta }_2,\mathrm{\dots },{\eta }_t\}$ of so-called fundamental units such that every unit $\epsilon$ of $R$ is a power (http://planetmath.org/GeneralAssociativity) product of these, multiplied by a root of unity:

$$\epsilon =\zeta \cdot {\eta }_1^{k_1}{\eta }_2^{k_2}\mathrm{\dots }{\eta }_t^{k_t}$$

Conversely, every such element $\epsilon$ of the field is a unit of $R$.

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 $H$ may be empty ($t=0$).  In the case of a single fundamental unit ($t=1$), which occurs e.g. in all real quadratic fields (http://planetmath.org/ImaginaryQuadraticField), there are two alternative units $\eta$ and its conjugate $\overline{\eta }$ which one can use as fundamental unit; then we can speak of the uniquely determined fundamental unit ${\eta }_1$ 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