If a field is set to be required, it is not possible to uncheck the Edit permission.

You can check whether the field is required by doing the following:
  1. Go to Setup.
  2. Under App Setup on the left, go to Customize -> <Object Type> -> Fields.
  3. Click the Field you wish to see the permissions of.
  4. Click View Field Accessibility.
  5. Find the Profile and Record type.  If Required is listed there, that means the field is required.