If I'm given a division algebra D with Z(D)=F, then how can I view D^{x} as an algebraic group defined over F? I'd like to see first how D^{x} can be given the structure of a variety defined over F, and then to see how the group law on D^{x} is defined over F.
Choose an Fbasis of D. The multiplication is described by certain quadratic functions, with respect to this basis; D* is given by the nonvanishing of a polynomial function (the norm). So the multiplication can be understood as defining an algebraic group structure on the complement of a hypersurface in an affine space. 


Suppose D splits over a finite extension K/F, i.e., the tensor product of D with K over F is isomorphic to M_{n}(K). Then D^{x} is the group of Fpoints of an algebraic group over F that exists as a direct factor (along with all other Fdivision algebras that split over K, and GL_{n,F}) in the restriction of scalars Res^{K}_{F} GL_{n,K}. I don't know an explicit presentation in general (say, starting from a Brauer class), although if K/F is a cyclic Galois extension, there is a nice cyclic algebra construction. I think more details can be found in Serre's Local Fields and CornellSilverman. 

