Is it necessary to install a cooler with Peltier Element?

Absolutely not necessarily!
The ideal application for peltier element is an instant drop in temperature within low ratio such as to get rid of cataract. Doctors use a soft contact on eye ball and then suddenly to make the thin membrane frozen, then to get rid of cataract together with the thin membrane. Its convenience is to have the operation done without hurting subtle eye structure. However, there are two limitations if apply the peltier element onto CPU:

1. Above 8 Watts ratio
2. Case space

The case space limits the dimension of heatsink to be small, so in this way peltier element itself plays as a thermal source. This way the heat can not be dissipated. A better solution in this case is to bring out the peltier element, and you might have a better cooling effect. If, the ratio of CPU is higher than how peltier element and heatsink can work together to cool, the peltier element actually bears extra burden keeping CPU at fixed temperature. When its performance is inefficient, it would burn the expensive peltier element since heatsink can't function and then your more expensive CPU will be burned. too.