A complete presentation of $(\mathbb{C}[X],\circ )$

We recall the fundamental theorem of J.F. Ritt, with a stress on the action of the affine group and canonical forms of complex polynomials. Then we give a complete presentation of the monoid $(\mathbb{C}[X],\circ )$. A list of decomposable polynomials is given in degrees $\le 12$.