When a AM radio wave reaches the antenna does the signal need to be in a closed circuit to be amplified?
When a AM radio wave reaches the antenna does the signal need to be in
  a closed circuit to be amplified?


Yes, and believe it or not there is a closed circuit. A simple monopole antenna uses ground as the return path - the incoming radio wave hits the antenna structure and a current circulates between monopole and ground and there will be an impedance too: -



The graph above shows what the electrical impedance of the monopole is and how it is dependant on the antenna length (height) and the wavelength of the radio wave. So, at about one quarter wave length the monopole looks purely resistive and that resistance is about 37 ohms (hard to see on the graph I understand). That's the impedance it presents to the rest of the circuit.

This means your radio wave is transformed into a signal with an output impedance of 37 ohms. But free space / air has an impedance too - it's \$120\pi\$ or about 377 ohms and this is due to the capacitance and inductance of free space i.e. the physical fundamental properties that dictate the speed of light.

So yes, there is a closed circuit. 

Here's an example - if you wanted to tune into an AM broadcast at 1 MHz you could construct a quarter wave monopole but, that monopole would be 75 metres long and present an impedance of 37 ohms. 

Or you could make a 15 metre long (0.05 wavelength) monopole that presents a capacitive impedance of about 1000 ohms (or 159 pF at 1 MHz). You would get more signal from the quarter wave antenna but, it would be really big and cumbersome then, to tune it you'd need a more complex circuit than the 15 metre antenna because that shorter antenna already looks like 159 pF and can directly connect to a coil to give good station selectivity. That's what the olde worlde crystal set users did.

Regarding your other question I have no idea what you mean so further information such as a circuit might be required.