Red,
There are too many unknown variables here but your have assumed certain things about use of the field and assumed that A-Hs will care about your privacy / safety. Your bad. Never underestimate self centred A-Hs. I thought this forum taught you that.
Talk to the local cops about the trespassing if you want but I think the results will be discouraging. If you are available 24/7 to document, report, follow up etc on the trespassing you might get somewhere but if there are numerous violators they can taunt you like hyenas.
You don't want to pay for a fence but that is the price of privacy. You can pay for a fence, security guards, paid duty cops, lawyers or thugs. Pick one or a combo.
Out of curiousity, how many feet of fence is required?
400' and a gate for the driveway