Your edit is starting to get it. If by the argument of the article x/y = cotton candy for all x,y, then probably the argument of the article isn't good. And the

Your edit is starting to get it. If by the argument of the article x/y = cotton candy for all x,y, then probably the argument of the article isn't good. And the reason is precisely that division in a field is taken to be nothing but a notational shorthand for multiplication by the multiplicative inverse.