So to be qualified to do aerial application work do you have to have a private license with a commercial rating?
Short answer: you can do aerial application on your own crops with a private certificate, but if you're going to do it commercially then you need a commercial certificate. In both cases you also need an agricultural aircraft operator certificate.
(b) Private operator—pilot. The applicant must hold a current U.S. private, commercial, or airline transport pilot certificate and be properly rated for the aircraft to be used.
(c) Commercial operator—pilots. The applicant must have available the services of at least one person who holds a current U.S. commercial or airline transport pilot certificate and who is properly rated for the aircraft to be used.
The FAA's AC 137, Certification Process for Agricultural Aircraft Operators has all the details about how to get an operator's certificate and the airman requirements are in section 2-3. It says:
Pilots designated to conduct operations for the commercial operator applicant must hold a current commercial or airline transport pilot certificate with the appropriate ratings
A private operator is limited to working his own crops:
Private agricultural aircraft operators may not conduct operations for compensation or hire, or conduct operations over a congested area. The operator must provide proof of property ownership or other property interest in the crop located on that property where the operation will be conducted.
By the way, there's no such thing as a "commercial rating" in the US; private and commercial are both certificates.
In the US, you are required to have a commercial license if you fly an aircraft "for compensation or hire". So yes a commercial license is required (in addition to lots of application specific training). I suppose you could dust your own crops without a commercial license depending on how the FAA determines how "compenstaion" applies in that case.
§61.133 Commercial pilot privileges and limitations. (a) Privileges—(1) General. A person who holds a commercial pilot certificate may act as pilot in command of an aircraft—
(i) Carrying persons or property for compensation or hire, provided the person is qualified in accordance with this part and with the applicable parts of this chapter that apply to the operation; and
(ii) For compensation or hire, provided the person is qualified in accordance with this part and with the applicable parts of this chapter that apply to the operation.