The 
th
exterior power 
of a vector space 
over a field 
is the degree-
component of its exterior
algebra 
,
where 
is a nonnegative integer. For 
, it is the vector space
span of the wedge products 
of vectors in 
, and 
(Stacks Project Authors 2026).
The 
th
exterior power of an element 
in an exterior algebra 
also denotes the wedge
product of 
with itself 
times. If 
with odd 
, then every power 
with 
is the zero element.
The situation for even degree elements
is different. For example, if
 |
(1)
|
then
 |  |  |  |  |
(2)
|
 |  |  |
(3)
| ||
 |  |  |
(4)
|
