To clarify some of the issues regarding airports and the National Airspace System, the FAA has classified the airspace into the following areas:
Class A: Is the Airspace including and above 18,000ft MSL. The most restricted airspace. Requires IFR rating and a clearance from Air Traffic Control (ATC) There are no airports in this class.
Class B: The biggest airports in the world. LAX, MIA, JFK, CYYZ, etc. Pilots must have a clearance, two way communication and there are aircraft equipment requirements.
Class C: Not as big as Class A, but airports with still a lot of traffic such as KSMF Sacramento, KCOS Colorado Springs, KALB Albany. Pilots must establish two way communication before entering and there are aircraft equipment requirements.
Class D: Smaller than Class C but still have an operational control tower. Pilots must establish and maintain two way communication.
Class B, C and D airports, have Control Towers and those are the people responsible for the airport. If you are within five miles of these airports you would need to call and get authorization before you could fly your Phantom anywhere near it. My guess is that considering the current status and opinion about drones around these airports, you would not get permission to fly a quadcopter within the 5 miles. These are the airport that show up as red circles in the DJI Maps.
Class E airspace is that which lies around Class A, B, C and D.
No operating towers. No two way communication required.
Class G is the left over airspace that is not within any of the prior mentioned classes. Definitely uncontrolled and not operating towers.
This website explains better the NAS:
http://www.vatusa.net/training/tiki-index.php?page=the_national_airspace_system
Most small airports are either city, county or state public airports or privately owned airports and airstrips. They are usually located in Class E or G for the most part but not necessarily.
These small airport do not show up in the DJI maps but according to the FAA they are still airports and you are still required to obtain permission. You would need to contact the owner of the airport and get permission to fly at or within the 5 miles of the center. There is a good chance that you could get permission to fly your Quadcopter in this class airports if you can convince them that you would not be a risk to the general aviation airplanes that use it.
A good website that has name and phone numbers for the owners of the airports is
www.airnav.com