The purposes of this work includes constructing an R-algebra consistent with our notions of geometry, and vector products; and then to do analysis on this construction to establish what field theory arises naturally within it. Next, extension of this construction toward a generalization of a linear form of the four-vector-Klein-Gordon equation is constructed and analyzed. Following that, further characteristics of the field are determined; and then the construction is extended into higher dimensions.The first chapter describes and establishes how a simple constructable R-algebra forms our elementary geometry, fashions our elementary vector constructs within it and erects it's products, and weaves the electromagnetic field expressed via Maxwell's equations; establishing geometry and physics as a mathematical model. The second chapter shows how a variation on the simple constructable flat-electromagnetic R-algebra of chapter one erects and weaves the gravitational field of Einstein's general relativity, further establishing geometry and physics as a mathematical model. In the third chapter, a further variation on the original R-algebra yields generalized Maxwell's equations with non-zero boson masses establishing an electromagnetic-nuclear field theory. Thus, this generalized algebra yields all the known fields of force establishing it as a full mathematical model of our four-dimensional conception of reality. The fourth chapter examines the electromagnetic-nuclear field equations developed last chapter, and determines some of their properties. The fifth chapter extends our original flat electromagnetic R-algebra beyond the four-dimensional experience. An eight-dimensional R-algebra is introduced and the cross-product for an R-algebra of types spacetime and NOT BOTH of any dimension is defined. A sixteen-dimensional R-algebra is introduced, and a generalization to an R-algebra of any dimension power of two.
