Do shareholders not elect a board of directors? Do the directors not have hire/fire authority over the executives? Not saying this chain of authority microman

Do shareholders not elect a board of directors? Do the directors not have hire/fire authority over the executives? Not saying this chain of authority micromanages every decision, but I suspect it's the rare CEO who goes out of his way to upset the board. Or is there some law that allows the CEO to say "You cannot fire me" (outside of anti-discrimination laws or such)?