Claudius: Is Hamlet mad?

I think that first of all, Claudius knows Hamlet does not like him. This is why he wants to know throughout the play if, in fact, Hamlet is mad. If Hamlet is, then he can easily dispense of him by sending him to England. This is why he acts so concerned-he is concerned, just not for Hamlet's sake. If Hamlet is not mad, then he could have something to worry about. Hamlet is coming to the age where he is the rightful age to inherit the throne. I think that Hamlet made a bad move in showing this play to the castle. It obviously shows that someone knows how King Hamlet was murdered. Hamlet invented the play, and so the obvious conclusion is that Hamlet knows the king was murdered. Claudius should now, more than ever, be concerned about Hamlet's state of mind.