  In this paper we study the commensurability of hyperbolic Coxeter groups of
finite covolume, providing three necessary conditions for commensurability.
Moreover we tackle different topics around the field of definition of a
hyperbolic Coxeter group: which possible fields can arise, how this field
determines a range of possible dihedral angles of a Coxeter polyhedron and we
provide two new sets of generators for quasi-arithmetic groups. This work is a
concise version of chapters 4 and 5 of the author's Ph.D. thesis.
