Don't you have the same rights as farmers?
To shot on sight of it coming onto your land?
I would have thought this, too.
TBH Midnight, Would have thought that you would find it very difficult to catch a dog that doesn't want to be caught - they're fast and they're agile. I also think that it would probably be unlikely to bite you - despite what yu might think by reading the papers, most dogs have a very strong inhibition when come to attacking people - but when they do, as others have said, you risk very serious injury.
Truly people-aggressive dogs, as opposed to dogs which are just trying to protect themselves and get out of an uncomfortable situation, will always target the face or throat of their victim. If you are lucky you could throw your arm up to protect yourself, but could end up maimed for life, especially if it is a large dog. Please don't consider this option.
However I can understand how desperate you must be even to consider doing something as risky as this. I think the police are shirking their duty - this is
a) a dog out of control in a public place
and
b) destruction of property (awful to describe the suffering of a living creature in such a way, but the law would see it as such, I think).
If you can, take a photograph of the dog (what breed is it? Is it something the police would have to take action over?)
If you know who it belongs to, tells the police and say that you want some action taking (I really can't understand their attitude - when a neighbour's dogs attacked mine - not for the first time - the police came out straight away and warned her to keep them under control). And if and when you find the owner, take them to court for your vet's bills.
While I don't think you would be able to catch the dog, I think you may be able to lure it with something very tasty - bacon, or cheese, perhaps, and put a slip lead on it. Then as someone else suggested, take it to the police station/ vet and find out who it belongs to, if you don't know already.
Alternatively - do you know anyone who would be prepared to lend you a stallion to put in your field? I understand that they can be as aggressive and as territorial as bulls, and wouldn't out up with a dog entering the field. (Of course, you'd have to ensure its safety).
This is a horrible position for you to be in. Animals in fields are so very vulnerable to dogs and idiots.