  Central in the Hopf algebra approach to the renormalization of perturbative
quantum field theory of Connes and Kreimer is their Algebraic Birkhoff
Decomposition. In this tutorial article, we introduce their decomposition and
prove it by the Atkinson Factorization in Rota-Baxter algebra. We then give
some applications of this decomposition in the study of divergent integrals and
multiple zeta values.
