# minimality of integral basis

The discriminant^{} $\mathrm{\Delta}:=\mathrm{\Delta}({\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{s})$ of the set $\{{\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{s}\}$ of integers of an algebraic number field^{} $K$ is a rational integer. If this set is an integral basis of $K$, then $|\mathrm{\Delta}|$ has the least possible (positive integer) value in the field $K$, and conversely. The value $d=\mathrm{\Delta}$ is equal for all integral bases of $K$, and it is called the discriminant or fundamental number of the field.

Title | minimality of integral basis |
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Canonical name | MinimalityOfIntegralBasis |

Date of creation | 2013-03-22 15:20:38 |

Last modified on | 2013-03-22 15:20:38 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 9 |

Author | Mathprof (13753) |

Entry type | Theorem |

Classification | msc 11R04 |

Related topic | CanonicalBasis |

Related topic | PropertiesOfDiscriminantInAlgebraicNumberField |

Defines | fundamental number |

Defines | discriminant of field |