If I'm given a division algebra D with Z(D)=F, then how can I view Dx as an algebraic group defined over F? I'd like to see first how Dx can be given the structure of a variety defined over F, and then to see how the group law on Dx is defined over F. 



Division Algebras as Algebraic Groups
Choose an F-basis 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. 


