minimality of integral basis

The discriminant  $\mathrm{\Delta }:=\mathrm{\Delta }({\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_s)$  of the set  $\{{\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_s\}$  of integers of an algebraic number field $K$ is a rational integer.  If this set is an integral basis of $K$, then $|\mathrm{\Delta }|$ has the least possible (positive integer) value in the field $K$, and conversely.  The value  $d=\mathrm{\Delta }$  is equal for all integral bases of $K$, and it is called the discriminant or fundamental number of the field.