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.

Title	minimality of integral basis
Canonical name	MinimalityOfIntegralBasis
Date of creation	2013-03-22 15:20:38
Last modified on	2013-03-22 15:20:38
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	9
Author	Mathprof (13753)
Entry type	Theorem
Classification	msc 11R04
Related topic	CanonicalBasis
Related topic	PropertiesOfDiscriminantInAlgebraicNumberField
Defines	fundamental number
Defines	discriminant of field