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# integral basis

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 $\mathbb{Q}$.

## Mathematics Subject Classification

11R04*no label found*

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