I'm trying to show that given a $1$-form $\omega$ and a $k$-form $\alpha$ such that $\alpha \wedge \omega = 0$ then there exists a $(k-1)$-form $\beta$ such that $\alpha = \omega \wedge \beta$.
I'm struggling to show this even in the case of multi-linear algebra and forgetting forms. I've been trying to use the fact that if $v_i$ are a basis for $V$ then $\{v_{i_1}\wedge...\wedge v_{i_k} : 1 \le i_1 < ... < i_k \le n\}$ is a basis for $\bigwedge ^k V$.
I thought perhaps we could extend $\omega$ (I'm talking just in the multilinear algebra case here and not thinking about manifolds - I'll worry about that after!) to a basis of $V$ and then $\alpha \wedge \omega = 0$ kills any terms containing $\omega$ in the expansion of $\alpha$ but I'm struggling to make any further progress,
Thanks for any help