# examples of ring of integers of a number field

###### Definition 1.

Let $K$ be a number field^{}. The ring of integers^{} of $K$, usually denoted by ${\mathrm{O}}_{K}$, is the set of all elements $\alpha \mathrm{\in}K$ which are roots of some monic polynomial with coefficients in $\mathrm{Z}$, i.e. those $\alpha \mathrm{\in}K$ which are integral over $\mathrm{Z}$. In other words, ${\mathrm{O}}_{K}$ is the integral closure of $\mathrm{Z}$ in $K$.

###### Example 1.

Notice that the only rational numbers which are roots of monic polynomials with integer coefficients are the integers themselves. Thus, the ring of integers of $\mathbb{Q}$ is $\mathbb{Z}$.

###### Example 2.

Let ${\mathcal{O}}_{K}$ denote the ring of integers of $K=\mathbb{Q}(\sqrt{d})$, where $d$ is a square-free integer. Then:

$${\mathcal{O}}_{K}\cong \{\begin{array}{cc}\mathbb{Z}\oplus \frac{1+\sqrt{d}}{2}\mathbb{Z},\text{if}d\equiv 1\mathrm{mod}\mathrm{\hspace{0.25em}4},\hfill & \\ \mathbb{Z}\oplus \sqrt{d}\mathbb{Z},\text{if}d\equiv 2,3\mathrm{mod}\mathrm{\hspace{0.25em}4}.\hfill & \end{array}$$ |

In other words, if we let

$$\alpha =\{\begin{array}{cc}\frac{1+\sqrt{d}}{2},\text{if}d\equiv 1\mathrm{mod}\mathrm{\hspace{0.25em}4},\hfill & \\ \sqrt{d},\text{if}d\equiv 2,3\mathrm{mod}\mathrm{\hspace{0.25em}4}.\hfill & \end{array}$$ |

then

$${\mathcal{O}}_{K}=\{n+m\alpha :n,m\in \mathbb{Z}\}.$$ |

###### Example 3.

Let $K=\mathbb{Q}({\zeta}_{n})$ be a cyclotomic extension of $\mathbb{Q}$, where ${\zeta}_{n}$ is a primitive $n$th root of unity^{}. Then the ring of integers of $K$ is ${\mathcal{O}}_{K}=\mathbb{Z}[{\zeta}_{n}]$, i.e.

$${\mathcal{O}}_{K}=\{{a}_{0}+{a}_{1}{\zeta}_{n}+{a}_{2}{\zeta}_{n}^{2}+\mathrm{\dots}+{a}_{n-1}{\zeta}_{n}^{n-1}:{a}_{i}\in \mathbb{Z}\}.$$ |

###### Example 4.

Let $\alpha $ be an algebraic integer^{} and let $K=\mathbb{Q}(\alpha )$. It is not true in general that ${\mathcal{O}}_{K}=\mathbb{Z}[\alpha ]$ (as we saw in Example $2$, for $d\equiv 1mod4$).

###### Example 5.

Let $p$ be a prime number^{} and let $F=\mathbb{Q}({\zeta}_{p})$ be a cyclotomic extension of $\mathbb{Q}$, where ${\zeta}_{p}$ is a primitive $p$th root of unity. Let ${F}^{+}$ be the maximal real subfield^{} of $F$. It can be shown that:

$${F}^{+}=\mathbb{Q}({\zeta}_{p}+{\zeta}_{p}^{-1}).$$ |

Moreover, it can also be shown that the ring of integers of ${F}^{+}$ is ${\mathcal{O}}_{{F}^{+}}=\mathbb{Z}[{\zeta}_{p}+{\zeta}_{p}^{-1}]$.

Title | examples of ring of integers of a number field |
---|---|

Canonical name | ExamplesOfRingOfIntegersOfANumberField |

Date of creation | 2013-03-22 15:08:09 |

Last modified on | 2013-03-22 15:08:09 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 7 |

Author | alozano (2414) |

Entry type | Example |

Classification | msc 13B22 |

Related topic | NumberField |

Related topic | AlgebraicNumberTheory |

Related topic | CanonicalBasis |

Related topic | IntegralBasisOfQuadraticField |