examples of ring of integers of a number fieldExamplesOfRingOfIntegersOfANumberField2005-03-15 15:59:392005-03-19 08:01:07Exampleintegral closure% this is the default PlanetMath preamble. as your knowledge
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% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Rats}{\mathbb{Q}}
\newcommand{\Gal}{\operatorname{Gal}}
\newcommand{\Cl}{\operatorname{Cl}}\begin{defn}
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 $\Ints$, i.e. those $\alpha\in K$ which are integral over $\Ints$. In other words, $\mathcal{O}_K$ is the integral closure of $\Ints$ in $K$.
\end{defn}
\begin{exa}
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 $\Rats$ is $\Ints$.
\end{exa}
\begin{exa}
Let $\mathcal{O}_K$ denote the ring of integers of $K=\Rats(\sqrt{d})$, where $d$ is a square-free integer. Then:
$$\mathcal{O}_K\cong \begin{cases}
\Ints\oplus \frac{1+\sqrt{d}}{2}\Ints, \text{ if } d\equiv 1 \ \operatorname{mod}\ 4,\\
\Ints\oplus \sqrt{d}\ \Ints, \text{ if } d\equiv 2,3
\operatorname{mod}\ 4. \end{cases}
$$
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
$$\mathcal{O}_K=\{ n+m\alpha : n,m \in \Ints \}.$$
\end{exa}
\begin{exa}
Let $K=\Rats(\zeta_n)$ be a cyclotomic extension of $\Rats$, where $\zeta_n$ is a primitive $n$th root of unity. Then the ring of integers of $K$ is $\mathcal{O}_K=\Ints[\zeta_n]$, i.e.
$$\mathcal{O}_K=\{ a_0 +a_1\zeta_n +a_2\zeta_n^2+\ldots+a_{n-1}\zeta_n^{n-1} : a_i \in \Ints\}.$$
\end{exa}
\begin{exa}
Let $\alpha$ be an algebraic integer and let $K=\Rats(\alpha)$. It is {\it not true in general} that $\mathcal{O}_K=\Ints[\alpha]$ (as we saw in Example $2$, for $d\equiv 1 \mod 4$).
\end{exa}
\begin{exa}
Let $p$ be a prime number and let $F=\Rats(\zeta_p)$ be a cyclotomic extension of $\Rats$, 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:
$$F^+=\Rats(\zeta_p+\zeta_p^{-1}).$$
Moreover, it can also be shown that the ring of integers of $F^+$ is $\mathcal{O}_{F^+}=\Ints[\zeta_p+\zeta_p^{-1}]$.
\end{exa}