Catalyzes the cyclization of farnesyl diphosphate to (E,E)-alpha-farnesene. (2E,6E)-farnesyl diphosphate = (3E,6E)-alpha-farnesene + diphosphate Binds 3 Mg(2+) ions per subunit. The Asp-Asp-Xaa-Xaa-Asp/Glu (DDXXD/E) motif is important for the catalytic activity, presumably through binding to Mg(2+). Belongs to the terpene synthase family.