  Arithmetic root systems are invariants of Nichols algebras of diagonal type
with a certain finiteness property. They can also be considered as
generalizations of ordinary root systems with rich structure and many new
examples. On the other hand, Nichols algebras are fundamental objects in the
construction of quantized enveloping algebras, in the noncommutative
differential geometry of quantum groups, and in the classification of pointed
Hopf algebras by the lifting method of Andruskiewitsch and Schneider. In the
present paper arithmetic root systems are classified in full generality. As a
byproduct many new finite dimensional pointed Hopf algebras are obtained.
  Key Words: Hopf algebra, Nichols algebra, Weyl groupoid
