Integral basis of a number field

An integral basis of a degree- $n$ number field $K$ is a basis over $\mathbb Z$ of its ring of integers: elements $\omega_1,\ldots,\omega_n$ such that each $x\in\mathcal O_K$ has a unique expression

x=m_1\omega_1+\cdots+m_n\omega_n,\qquad m_i\in\mathbb Z.

Existence and distinction

Every number field has an integral basis. Its elements are also a rational basis of $K$ , but the converse fails even for a rational basis made of algebraic integers.

For example, $(1,\sqrt{-3})$ is a rational basis of $\mathbb Q(\sqrt{-3})$ but is not integral: it misses $(1+\sqrt{-3})/2$ . The pair $(1,(1+\sqrt{-3})/2)$ is an integral basis.

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, §2, Proposition 2.29 and Definition 2.32.