S-forms of a Finitely Generated Field Extension

Abstract

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.