The kth exterior power of an element α in an exterior algebra Λ V is given by the wedge product of α with itself k times. Note that if α has odd degree, then any higher power of α must be zero. The situation for even degree forms is different. For example, if α = e_1 ⋀e_2 + e_3 ⋀e_4 + e_5 ⋀e_6, then α^2 | = | 2e_1 ⋀e_2 ⋀e_3 ⋀e_4 + 2e_1 ⋀e_2 ⋀e_5 ⋀e_6 + 2e_3 ⋀e_4 ⋀e_5 ⋀e_6 α^3 | = | 6e_1 ⋀e_2 ⋀e_3 ⋀e_4 ⋀e_5 ⋀e_6 α^4 | = | 0.

exterior algebra | wedge product