Integral Basis for Number Field

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:

Theorem

If $\{{\alpha }_1,\dots ,{\alpha }_n\}$ 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

$$x=m_1{\alpha }_1+\cdots +m_n{\alpha }_n\in \mathbb{K}$$

then $x\in {\mathcal{O}}_{\mathbb{K}}$ if and only if all $m_i$ are in $\mathbb{Z}$.