# power basis over $\mathbb{Z}$

Let $K$ be a number field with $[K:\mathbb{Q}]=n$ and ${\mathcal{O}}_{K}$ denote the ring of integers^{} of $K$. Then ${\mathcal{O}}_{K}$ has a power basis over $\mathbb{Z}$ (sometimes shortened simply to power basis) if there exists $\alpha \in K$ such that the set $\{1,\alpha ,\mathrm{\dots},{\alpha}^{n-1}\}$ is an integral basis for ${\mathcal{O}}_{K}$. An equivalent^{} (http://planetmath.org/Equivalent3) condition is that ${\mathcal{O}}_{K}=\mathbb{Z}[\alpha ]$. Note that if such an $\alpha $ exists, then $\alpha \in {\mathcal{O}}_{K}$ and $K=\mathbb{Q}(\alpha )$.

Not all rings of integers have power bases. (See the entry biquadratic field for more details.) On the other hand, any ring of integers of a quadratic field has a power basis over $\mathbb{Z}$, as does any ring of integers of a cyclotomic field^{}. (See the entry examples of ring of integers of a number field for more details.)

Title | power basis over $\mathbb{Z}$ |
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Canonical name | PowerBasisOvermathbbZ |

Date of creation | 2013-03-22 15:56:55 |

Last modified on | 2013-03-22 15:56:55 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 17 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 11R04 |

Synonym | power basis |

Synonym | power bases |

Related topic | ConditionForPowerBasis |