Necessary and sufficient conditions for $\alpha \wedge \beta = 0$

Question

given $\alpha \in \mathcal{A}^1(V)$ and $\beta \in \mathcal{A}^k(V)$, we can conclude that if there's a $\gamma \in \mathcal{A}^{k-1}(V)$ s.t. $\beta = \alpha \wedge \gamma$, then $\alpha \wedge \beta = \alpha \wedge \alpha \wedge \gamma = 0$ since $\alpha$ is an odd valued tensor.

Now, given $\alpha \wedge \beta = 0$, how can I prove existence of $\gamma \in \mathcal{A}^{k-1}(V)$ s.t. $\beta = \alpha \wedge \gamma$?

Thanks!