The number field discriminant of a number field 
of extension field degree ![n=[K:Q]](/images/equations/NumberFieldDiscriminant/Inline2.svg)
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 a number
field order 
,
the analogous determinant using a 
-basis of 
satisfies
![D_O=[O_K:O]^2D_K.](/images/equations/NumberFieldDiscriminant/NumberedEquation2.svg) |
The absolute value of 
is the absolute discriminant of 
. For example, 
has integral basis
1 and 
and discriminant 8 (Milne 2020, p. 39). Order discriminants are arithmetic size
parameters for packets of periodic torus orbits
(Einsiedler et al. 2011, pp. 823-824).
The discriminant of the number field 
generated by an algebraic
number 
is computed in the Wolfram Language
as NumberFieldDiscriminant[a].
