I know how zombies can be defeated, because I've been playing the game "Oblivion" (a great sword and sorcery game).

You get a flame bow. That's an enchanted bow that shoots fiery arrows. Zombies don't like fire. You put a couple of fiery arrows in that zombie, and he's kaput. A modern flamethrower or any other source of conflagration will do the job fine too, if you can't get an enchanted bow.