The reduced norm of a finite-dimensional central simple algebra 
over a field 
is a map 
obtained from a determinant
after extending scalars. Here central simple means that 
is an associative algebra with identity, has center 
, and has no proper nonzero two-sided ideals.
If 
, then over an algebraic
closure 
there is an isomorphism 
, where
is the matrix
ring of 
matrices. The reduced norm of 
is the determinant of
its image in this matrix ring. It belongs to 
and is independent of the chosen isomorphism
(Gille and Szamuely 2006, pp. 37-38).
The reduced norm is multiplicative, and 
for a scalar 
. For 
it is the ordinary determinant.
In a central division algebra, the elements of
reduced norm one form a subgroup of the multiplicative
group of nonzero elements.
