# fundamental units

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 such that every unit $\varepsilon$ of $R$ is a power (http://planetmath.org/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,  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 FundamentalUnits 2014-11-24 16:38:36 2014-11-24 16:38:36 pahio (2872) pahio (2872) 22 pahio (2872) Definition msc 11R27 msc 11R04 NumberField AlgebraicInteger