Let $K$ be a number field. A set of algebraic integers $\{\alpha_1,\ldots,\alpha_s\}$ is said to be an integral basis for $K$ if every $\gamma$ in $\mathcal{O}_K$ can be represented uniquely as an integer linear combination of $\{\alpha_1,\ldots,\alpha_s\}$ (i.e. one can write $\gamma = m_1 \alpha_1 + \cdots + m_s \alpha_s$ with $m_1,\ldots,m_s$ (rational) integers).

If $I$ is an ideal of $\mathcal{O}_K$ then $\{\alpha_1,\ldots,\alpha_s\} \in I$ is said to be an integral basis for $I$ if every element of $I$ can be represented uniquely as an integer linear combination of $\{\alpha_1,\ldots,\alpha_s\}$

(In the above, $\mathcal{O}_K$ denotes the ring of algebraic integers of $K$ )

An integral basis for $K$ over $\mathbb{Q}$ is a basis for $K$ over $\mathbb{Q}$