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It is the purpose of this paper to find necessary and sufficient conditions for an exterior p-form zu to factor into a product ZL’ = t A y1 h yz A ... A yTc of k l-forms, and a (p K)-f orm; and to establish a maximum fork (Theorem 1). This is done by introducing a certain invariant for w called “length” which is subsequently proved to be equal to the “rank” of the dual (E p)vector (Theorem 2). An interesting application is that an (212 I)-form on an (2% + l)dimensional vector-space always f;ictors into a product of a l-form and a (272 2)-form (Corollary 2.1). One also obtains a quick proof for the standard theorem that an exterior p-form w is fully-decomposable (i.e=, a product of l-forms) if and only if it is of minimal rankp, and that p + 2 < Y(ZL’) < n when ‘W is not fully decomposable (Corollary 2.2).

Comparing the dimensions of the two spaces S p (E) and S n–p (E) we see that these spaces are isomorphic.