# 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 $\mathcal{O}_{K}$, is the set of all elements $\alpha\in K$ which are roots of some monic polynomial with coefficients in $\mathbb{Z}$, i.e. those $\alpha\in K$ which are integral over $\mathbb{Z}$. In other words, $\mathcal{O}_{K}$ is the integral closure of $\mathbb{Z}$ 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:

 $\mathcal{O}_{K}\cong\begin{cases}\mathbb{Z}\oplus\frac{1+\sqrt{d}}{2}\mathbb{Z% },\text{ if }d\equiv 1\ \operatorname{mod}\ 4,\\ \mathbb{Z}\oplus\sqrt{d}\ \mathbb{Z},\text{ if }d\equiv 2,3\operatorname{mod}% \ 4.\end{cases}$

In other words, if we let

 $\alpha=\begin{cases}\frac{1+\sqrt{d}}{2},\text{ if }d\equiv 1\ \operatorname{% mod}\ 4,\\ \sqrt{d},\text{ if }d\equiv 2,3\operatorname{mod}\ 4.\end{cases}$

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.

 $\mathcal{O}_{K}=\{a_{0}+a_{1}\zeta_{n}+a_{2}\zeta_{n}^{2}+\ldots+a_{n-1}\zeta_% {n}^{n-1}:a_{i}\in\mathbb{Z}\}.$
###### 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 $d\equiv 1\mod 4$).

###### 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 ExamplesOfRingOfIntegersOfANumberField 2013-03-22 15:08:09 2013-03-22 15:08:09 alozano (2414) alozano (2414) 7 alozano (2414) Example msc 13B22 NumberField AlgebraicNumberTheory CanonicalBasis IntegralBasisOfQuadraticField