I'm pretty sure you don't want a soil sterilent, that is pretty darned extreame. Talk to a local farm store/ fertilizer plant, they will be able to direct you the right direction. A couple years of spraying with something as simple as 2,4-d will end your problems most likely. The 2,4-D is nearly free as far as chemical goes, the application will likely cost the most because it'll take longer to mix than to spray it on.
Don't go and plow the damned things under cause burying the problem will give you a bank vault of the boogers in the soil. Right now they are on top, and as they germinate you can kill them and be done, The key is not to let a single one go to seed for a few years, you can wear them out. If you can mow and spray the area like a lawn so much the better, a lawn care outfit may be an easier route for you to go. Find an outfit doing a large appt complex or industrial site they will have large enough sprayer to help you out. I would plan on three sprayings a year. Learn what they look like when under 2" tall, walk the field once every two weeks looking for the things and spray while they are small.