fundamental units

The ring $R$ of algebraic integers of any algebraic numberfield contains a finite set$H=\{{\eta }_1,{\eta }_2,\mathrm{\dots },{\eta }_t\}$ of so-calledfundamental units such that every unit $\epsilon$ of$R$ is a power (http://planetmath.org/GeneralAssociativity) product ofthese, 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 aunit of $R$.

Examples:  units of quadratic fields,  units of certain cubic fields (http://planetmath.org/UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding)

For some algebraic number fields, such as all imaginaryquadratic fields, the set $H$ may be empty ($t=0$).  In thecase of a single fundamental unit ($t=1$), which occurs e.g.in allreal quadratic fields (http://planetmath.org/ImaginaryQuadraticField),there are two alternative units$\eta$ and its conjugate $\overline{\eta }$ which one can use asfundamental unit; then we can speak of the uniquelydetermined fundamental unit ${\eta }_1$ which is greater than 1.