Denominator of Algebraic Number

Definition

The denominator of an algebraic number \(\alpha\) is the smallest positive integer \(k\) such that \(k \alpha\) is an algebraic integer.

A priori, this so called denominator may not exist. As such, the remainder of this note is devoted to proving that indeed, as above, it is well defined.

Theorem

For any element \(\alpha\) of a number field \(\mathbb{K}\), there exists an integer \(k \in \mathbb{Z} \setminus \{0\}\) such that \(k\alpha \in \mathcal{O}_\mathbb{K}\).

Proof

Let \(\mathbb{K}\) be a number field of degree \(n\) over \(\mathbb{Q}\). \(\mathbb{K}\) has an integral basis \(\{\alpha_1, \dots, \alpha_n\}\) and as such, every element \(\alpha \in \mathbb{K}\) can be expressed as

\[ \alpha = \frac{m_1}{k_1}\alpha_1 + \frac{m_2}{k_2}\alpha_2 + \dots + \frac{m_n}{k_n}\alpha_n\]

where \(m_1, \dots, m_n, k_1, \dots, k_n \in \mathbb{Z}\) and \(k_1, \dots, k_n \neq 0\). That is, everything is a \(\mathbb{Q}\) linear combination of the basis. Then, letting \(k = \mathrm{lcm}(k_1, \dots, k_n)\), we have that

\[ k\alpha = m_1\frac{k}{k_1}\alpha_1 + \dots + m_n\frac{k}{k_n}\alpha_n\]

is a \(\mathbb{Z}\) linear combination of the integral basis and hence an algebraic integer. That is, \(k \alpha \in \mathcal{O}_\mathbb{K}\).

Corollary

For any algebraic number \(\alpha\), there exists a positive integer \(k\) such that \(k \alpha\) is an algebraic integer.

Proof

Of course every algebraic number \(\alpha\) is an element of the number field \(\mathbb{Q}(\alpha)\) and thus there exists a positive integer \(k\) such that \(k \alpha \in \mathcal{O}_{\mathbb{Q}(\alpha)}\) which is a subset of the set of all algebraic integers.