PWM control of Peltier with Arduino+Thermistor
Your info is basically correct. 

Unlike motor, Peltier needs no flyback diode as it does not generate harmful back EMF.

Low side switching means connection is Batt_pos, load, D of MOSFET and S of MOSFET to Batt_Neg. Send signal High to G for on. Low for off.  

To build from discrete MOSFET, I do not have full circuit on hands, but, you should get tons of tutorial from web. 

Full H bridge in this diagram is for hot/cold control. For hot-only, your example is fine.