# minimality of integral basis

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 discriminant or fundamental number of the field.

Title minimality of integral basis MinimalityOfIntegralBasis 2013-03-22 15:20:38 2013-03-22 15:20:38 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Theorem msc 11R04 CanonicalBasis PropertiesOfDiscriminantInAlgebraicNumberField fundamental number discriminant of field