  The concept of arithmetic root systems is introduced. It is shown that there
is a one-to-one correspondence between arithmetic root systems and Nichols
algebras of diagonal type having a finite set of (restricted)
Poincare'-Birkhoff-Witt generators. This has strong consequences for both
objects. As an application all rank 2 Nichols algebras of diagonal type having
a finite set of (restricted) Poincare'-Birkhoff-Witt generators are determined.
  Key Words: Brandt groupoid, Hopf algebra, pseudo-reflections, Weyl group
