# integral basis

Let $K$ be a number field. A set of algebraic integers^{}
$\{{\alpha}_{1},\mathrm{\dots},{\alpha}_{s}\}$ is said to be an integral basis for $K$
if every $\gamma $ in ${\mathcal{O}}_{K}$ can be represented uniquely
as an integer linear combination^{} of $\{{\alpha}_{1},\mathrm{\dots},{\alpha}_{s}\}$ (i.e. one can write $\gamma ={m}_{1}{\alpha}_{1}+\mathrm{\cdots}+{m}_{s}{\alpha}_{s}$ with ${m}_{1},\mathrm{\dots},{m}_{s}$
(rational) integers).

If $I$ is an ideal of ${\mathcal{O}}_{K}$, then $\{{\alpha}_{1},\mathrm{\dots},{\alpha}_{s}\}\in I$ is said to be an integral basis for $I$ if every element of $I$ can be represented uniquely as an integer linear combination of $\{{\alpha}_{1},\mathrm{\dots},{\alpha}_{s}\}$.

(In the above, ${\mathcal{O}}_{K}$ denotes the ring of algebraic integers of $K$.)

An integral basis for $K$ over $\mathbb{Q}$ is a basis for $K$ over $\mathbb{Q}$.

Title | integral basis |

Canonical name | IntegralBasis |

Date of creation | 2013-03-22 12:36:03 |

Last modified on | 2013-03-22 12:36:03 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 12 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 11R04 |

Synonym | minimal basis |

Synonym | minimal bases |

Related topic | AlgebraicInteger |

Related topic | Integral |

Related topic | Basis |

Related topic | DiscriminantOfANumberField |

Related topic | ConditionForPowerBasis |

Related topic | BasisOfIdealInAlgebraicNumberField |

Related topic | CanonicalFormOfElementOfNumberField |

Defines | integral bases |