Hilbert's Third Problem: Scissors Congruence

additive affine Archimedean ordered assertion assume basic i-fold cylinder canonical decomposition codimensional 1 face cohomology commutative comodule Coxeter Coxeter group defined definite inner product Dehn invariant denote dihedral angle dimension direct sum elements Euclidean exterior normal filtration finite number follows formula function G-scissors geometric n-simplex graded volume map Hadwiger invariants Hilbert's third problem Hopf algebra hyperbolic identified induction injective inner product integer isomorphic Jessen k-algebra k-basis k-linear Lemma Letk Lsp(A Lsp(A(I Minkowski sum multiplicative n-simplices nonzero normal ordered field orthogonal complement polyhedra polyhedron positive definite inner preceding Proj proof Proposition 2.2 PS/CS respect to simple result root closed field scissors congruence scissors congruence data scissors congruence problem separate the points shows simple subdivision simplex spherical n-simplex square root closed St(k subgroup subset subspace symmetric algebra third problem Tor(PS torsion unique unit vector ΕΦ