fundamental unitsFundamentalUnits2004-08-06 12:32:222006-10-16 09:54:43Definitiongroup of unitsDirichlet's unit theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe 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 {\em fundamental units} such that every unit $\varepsilon$ of $R$ is a \PMlinkname{power}{GeneralAssociativity} 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 $R$.
Examples:\, units of quadratic fields,\, \PMlinkname{units of certain cubic fields}{UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding}
For some algebraic number fields, such as all imaginary \PMlinkname{quadratic fields}{QuadraticNumberField}, 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 {\em the} uniquely determined fundamental unit $\eta_1$ which is greater than 1.