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[parent] examples of ring of integers of a number field (Example)
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:
$\displaystyle \mathcal{O}_K\cong \begin{cases} \mathbb{Z}\oplus \frac{1+\sqrt{d... ...sqrt{d}\ \mathbb{Z}, \text{ if } d\equiv 2,3 \operatorname{mod}\ 4. \end{cases}$
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 $ n$th root of unity. Then the ring of integers of $ K$ 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 $ 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:
$\displaystyle F^+=\mathbb{Q}(\zeta_p+\zeta_p^{-1}).$
Moreover, it can also be shown that the ring of integers of $ F^+$ is $ \mathcal{O}_{F^+}=\mathbb{Z}[\zeta_p+\zeta_p^{-1}]$.



"examples of ring of integers of a number field" is owned by alozano.
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See Also: number field, algebraic number theory, canonical basis, integral basis of quadratic field


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Cross-references: maximal real subfield, prime number, algebraic integer, root of unity, primitive, cyclotomic extension, square-free, integer, rational numbers, integral closure, integral, coefficients, monic polynomial, roots, ring of integers, number field
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This is version 4 of examples of ring of integers of a number field, born on 2005-03-15, modified 2005-03-19.
Object id is 6879, canonical name is ExamplesOfRingOfIntegersOfANumberField.
Accessed 2567 times total.

Classification:
AMS MSC: 13B22 (Commutative rings and algebras :: Ring extensions and related topics :: Integral closure of rings and ideals ; integrally closed rings, related rings )
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