# Division Algebras as Algebraic Groups

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.

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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.

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And for the record, this is the exact same way that you show GL_n is an algebraic group. –  Tyler Lawson Oct 28 '09 at 13:01
Is it part of the general theory of division algebras that the norm is a polynomial function? –  Joel Dodge Oct 28 '09 at 19:27
@Joel: After base change to a matrix ring, the norm becomes the determinant. –  S. Carnahan Oct 29 '09 at 2:58