To provide an α-olefin polymer having high stereoregularity and a broad molecular weight distribution in high polymerization activity.

There is provided a process for polymerizing an α-olefin comprising polymerizing or copolymerizing an α-olefin in the presence of a catalyst comprising (A) a solid catalyst component essentially consisting of magnesium, titanium, a halogen, and an electron donor; (B) an organoaluminum compound component; (C) di(cyclopentyl) dimethoxysilane, and (D) a specified polycyclic aminosilane.