My blue inductor is missing..
Is it possible to just replace the blue inductor (since it is missing) with some solder? Will this work? I have tried joining the joints with a screw driver but nothing happens... Still no service. Or will I have to replace it with another iPhone 4 blue inductor?
The answer to your questions, how to fix the blue inductor coil causing a signal problem, the solution is easy we have repaired many with a 100% success rate. If you look at the signal antenna which you plug the cable into, simply run a wire from the left side bridge it to the right side point on the blue coil and follow it all the way to the furthest (right) component on the top left side. This will act as a 'bypass route' and your signal will start to work again.
All it takes is a bit of micro-soldering knowledge and some 0.1mm wire and basing soldering equipment. This will fix your problem guaranteed, providing it was the missing blue coil that caused your signal issue in the first place.