The number field discriminant of a number field 
is the integer
 |
where 
,
..., 
form an integral basis of the ring
of integers of 
and the trace is the field trace.
Changing the integral basis leaves this nonzero
integer unchanged. For an order 
, the analogous determinant using a basis of 
satisfies
![D_O=[O_K:O]^2D_K.](/images/equations/NumberFieldDiscriminant/NumberedEquation2.svg) |
Its absolute value is the absolute discriminant. For example, 
has integral basis 1 and 
and discriminant 8 (Milne 2020). Order discriminants
are arithmetic size parameters for periodic torus packets.
The discriminant of the number field 
generated by an algebraic
number 
is computed in the Wolfram Language
as NumberFieldDiscriminant[a].
