Section 4 contains a list of all compact hyperbolic Coxeter polytopes the combinatorial type of which is the product of two simplices of dimension greater 1. Together with results of Kaplinskaja this completes the classification of compact hyperbolic $d$-polytopes with $d+2$ facets.
In section 5 it is proved that compact hyperbolic Coxeter $d$-polytopes with $d+3$ facets exist only for $d \leq 8$ and that the example of Bugaenko for $d = 8$ is unique.
Due to a result of Nikulin, the degree of the ground field of arithmetic Coxeter $d$-polytopes, $d \geq 10$, is bounded. In section 6 this is shown to be true also for $d =9$ and for all $d$-polytopes bounded by at most $d+4$ facets.
The last section provides examples of arithmetic Coxeter polytopes with ground fields ${\Bbb Q}(\sqrt{13})$, ${\Bbb Q}(\sqrt{17})$ and ${\Bbb Q}(\sqrt {21})$. }
Key words and phrases: compact hyperbolic Coxeter polytope, hyperbolic reflection group, arithmetic reflection group, ground field of an arithmetic reflection group.
1991 Mathematics Subject Classification: Primary 20H15; Secondary 57S30, 51F15.
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Contact: Frank.Esselmann@Mathematik.Uni-Dortmund.DE