# 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 |