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minimality of integral basis

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\begin{document}
The discriminant\, $\Delta := \Delta(\alpha_1,\,\alpha_2,\,\ldots,\,\alpha_s)$\, of the set\, $\{\alpha_1,\,\alpha_2,\,\ldots,\,\alpha_s\}$\, of integers of an algebraic number field $K$ is a rational integer.\, If this set is an integral basis of $K$, then $|\Delta|$ has the least possible (positive integer) value in the field $K$, and conversely.\, The value\, $d = \Delta$\, is equal for all integral bases of $K$, and it is called the {\em discriminant} or {\em fundamental number} of the field.
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