algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety multiplicative structure; skew fields over number fields; Hasse; norm principle; algebraic group; group of rational points; quadratic forms; Skolem-Noether theorem; algebra of quaternions; class field theory; direct subgroup; Spin(f); SL(1,D); trace 