Let $K$ be a finitely generated extension of a field $k$ of characteristic $p\not =0$. By means of exponents of $K/k$, we introduce the notion of $s$-forms ($s$ being a positive integer less than or equal to $insep(K/k)$ of $K/k$ as a natural generalization of forms of $K/k$. In light of results obtained by James K. Deveney and John N. Mordeson in their investigation on the forms of a finitely generated field extension [Deveney and Mordeson: Can. J. Math. 31(3), 655–662 (1979)], necessary and sufficient conditions characterizing $s$-forms of $K/k$ are given allowing in particular the existence of a unique minimal $s$-form (irreducible $s$-form) of $K/k$ and, accordingly, the development of properties of irreducible $s$-forms of $K/k$. We also seek to identify possible relationships between the structure and invariants of $K/k$ and those of its irreducible $s$-form.