You can search the USPTO,
http://www.uspto.gov/ (you'd think there'd be another s in that URL, right?) Note what years may not be there, though and there are submarine patents that are in process but haven't surfaced yet.
Improvements to another's invention can be patented.
Must be "non-obvious"; but what that means is always disputed. If a great many of people have seen the problem, but not had your idea, that's good evidence it's not obvious. Must not have been published more than two years ago by you, or anywhere by someone else (unless you can show that's your work they just published.) Can't be prior art (already being used), anywhere.
Meet the above conditions, and the first to the patent office with the invention, a roll of cash, and someone - anyone - willing to say that, yup, um, yeah they totally thought that up, wins! Didn't used to be that way but it is now.