Quillen and André have rigorized and explored a notion of cohomology algebras or, more generally, simplicial commutative algebras. They were able to do a number of systematic calculations, especially when concerned with a local ring with resiude field of characteristic 0, but the case when the characteristic was non-zero remained a problem. However, for certain applications – for example, to homotopy theory – the non-zero characteristic case is vital. In this paper we explore André-Quillen cohomology of supplemented algebras avec the field $\mathbb {F}_2$ of two elements, and completely determine the structure of this cohomology, including a product and “Steenrod” operations. A necessary part of the program is a complete examination of the homotopy theory of simplicial algebras. For this we draw on the work of many authors.