  We give an explicit and essentially minimal list of defining relations of a
Nichols algebra of diagonal type with finite root system. This list contains
the well-known quantum Serre relations but also many new variations. A
conjecture by Andruskiewitsch and Schneider states that any finite-dimensional
pointed Hopf algebra over an algebraically closed field of characteristic zero
is generated as an algebra by its group-like and skew-primitive elements. As an
application of our main result, we prove the conjecture when the group of
group-like elements is abelian.
