2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170005We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules.37 pages; exposition improved and more detail given; Lemma 3.1 strengthenedAlgebraic GeometryRings and Algebras12E30; 14Htext