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Examples Of Ring Of Integers Of A Number Field

Definition 1 Let be a number field. The ring of integers of , 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 .

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

is a square-free integer. Then:

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

In other words, if we let

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

then

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

Example 3 Let $K=\mathbb{Q}(\zeta_n)$ be a cyclotomic extension of $\mathbb{Q}$ , where $\zeta_n$ is a primitive

th root of unity. Then the ring of integers of

is $\mathcal{O}_K=\mathbb{Z}[\zeta_n]$ , i.e.

$\displaystyle \mathcal{O}_K=\{ a_0 +a_1\zeta_n +a_2\zeta_n^2+\ldots+a_{n-1}\zeta_n^{n-1} : a_i \in \mathbb{Z}\}.$

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

, for $d\equiv 1 \mod 4$ ).

Example 5 Let

be a prime number and let $F=\mathbb{Q}(\zeta_p)$ be a cyclotomic extension of $\mathbb{Q}$ , where $\zeta_p$ is a primitive

th root of unity. Let

be the maximal real subfield of

. It can be shown that:

$\displaystyle F^+=\mathbb{Q}(\zeta_p+\zeta_p^{-1}).$

Moreover, it can also be shown that the ring of integers of

is $\mathcal{O}_{F^+}=\mathbb{Z}[\zeta_p+\zeta_p^{-1}]$ .

1	Liber Abaci
1	Basic Polynomial
1	Integer
1	Integral
1	Integral Closure
1	Prime
2	Root Of Unity
2	Number Field
2	Completely Simple Semigroup
3	Root Of A Tree
3	Square Free Number
4	Monic
5	Algebraic Integer
6	Cyclotomic Extension
7	Totally Real And Imaginary Fields