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The ring $R$ of algebraic integers of any algebraic number
field contains a finite set
$H=\{\eta_{1},\,\eta_{2},\,\ldots,\,\eta_{t}\}$ of so-called
fundamental units such that every unit $\varepsilon$ of
$R$ is a power product^{} of
these, multiplied by a root of unity:

$\varepsilon=\zeta\!\cdot\!\eta_{1}^{{k_{1}}}\eta_{2}^{{k_{2}}}\ldots\eta_{t}^{% {k_{t}}}$ |

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

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, 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.

## Mathematics Subject Classification

11R27*no label found*11R04

*no label found*

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