Integral Basis for Number Field
A collection of elements \(\{\alpha_1, \dots, \alpha_n\}\) of a number field \(\mathbb{K} \supseteq \mathbb{Q}\) is called an integral basis if every element of the ring of integers \(\mathcal{O}_\mathbb{K}\) can be expressed as a \(\mathbb{Z}\) linear combinations of the set. That is for every \(x \in \mathcal{O}_\mathbb{K}\), there exists integers \(m_1, \dots, m_n\) such that
Sometimes we call this an integral basis for \(\mathbb{K}\) or an integral basis for \(\mathcal{O}_\mathbb{K}\), both of which mean the same thing.
A very important characterisation of an integral basis is the following result:
TheoremIf \(\{\alpha_1, \dots, \alpha_n\}\) is an integral basis of \(\mathcal{O}_\mathbb{K}\), then \(\{\alpha_1, \dots, \alpha_n\}\) is a basis for \(\mathbb{K}\) over \(\mathbb{Q}\). Moreover, if
\[ x = m_1\alpha_1 + \dots + m_n\alpha_n \in \mathbb{K}\]then \(x \in \mathcal{O}_\mathbb{K}\) if and only if all \(m_i\) are in \(\mathbb{Z}\).
