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 $\Ints$ , i.e. those $\alpha\in K$ which are integral over $\Ints$ . In other words, $\mathcal{O}_K$ is the integral closure of $\Ints$ in $K$ .

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 $\Rats$ is $\Ints$ .

Example 2   Let $\mathcal{O}_K$ denote the ring of integers of $K=\Rats(\sqrt{d})$ , where $d$ is a square-free integer. Then:

In other words, if we let

then $$\mathcal{O}_K=\{ n+m\alpha : n,m \in \Ints \}.$$

Example 3   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. $$\mathcal{O}_K=\{ a_0 +a_1\zeta_n +a_2\zeta_n^2+\ldots+a_{n-1}\zeta_n^{n-1} : a_i \in \Ints\}.$$

Example 4   Let $\alpha$ be an algebraic integer and let $K=\Rats(\alpha)$ . It is not true in general that $\mathcal{O}_K=\Ints[\alpha]$ (as we saw in Example $2$ , for $d\equiv 1 \mod 4$ ).

Example 5   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 $\mathcal{O}_{F^+}=\Ints[\zeta_p+\zeta_p^{-1}]$ .