What is a ring of integers? What does “integral over \(\ZZ\)” mean?
Solution
Let \(K\) be a number field, i.e. a finite extension of \(\QQ\).
An element \(\alpha\in K\) is integral over \(\ZZ\) if it satisfies a monic polynomial \[ \alpha^n+a_{n-1}\alpha^{n-1}+\cdots+a_0=0 \] with coefficients \(a_i\in\ZZ\).
The ring of integers of \(K\) is \[ \mathcal O_K = \{\alpha\in K:\alpha\text{ is integral over }\ZZ\}. \]
1. The set \(\mathcal O_K\) is a subring of \(K\) containing \(\ZZ\).
The elements integral over a ring form a subring of any overring: sums, differences, and products of integral elements are integral. Every integer \(m\) is integral because it is a root of the monic polynomial \(x-m\).
2. For \(K=\QQ\), one has\[ \mathcal O_{\QQ}=\ZZ. \]
Let \(a/b\in\QQ\) be in lowest terms and integral over \(\ZZ\). If it satisfies a monic polynomial with integer coefficients, multiplying by \(b^n\) shows that \(b\mid a^n\). Coprimality forces \(b=1\), so the rational number is an integer.
Thus \(\mathcal O_K\) is the integral closure of \(\ZZ\) in the number field \(K\).