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

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

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

$${𝒪}_K=\{n+m\alpha :n,m\in \mathbb{Z}\}.$$

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 ${𝒪}_K=\mathbb{Z}[{\zeta }_n]$, i.e.

$${𝒪}_K=\{a_0+a_1{\zeta }_n+a_2{\zeta }_n^2+\mathrm{\dots }+a_{n-1}{\zeta }_n^{n-1}:a_i\in \mathbb{Z}\}.$$

Example 4.

Let $\alpha$ be an algebraic integer and let $K=\mathbb{Q}(\alpha )$. It is not true in general that ${𝒪}_K=\mathbb{Z}[\alpha ]$ (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 ${𝒪}_{F^{+}}=\mathbb{Z}[{\zeta }_p+{\zeta }_p^{-1}]$.