^ Normally you'd make it really long and then tune it with a (believe it or not ) long wire antenna tuner. I believe MFJ still sells a fairly economical one, not hard to make one just for receiving. A long dipole is a nice not-too-difficult project. Also consider a fancy loop antenna. What is great about them is, besides being reasonably small (compared to LW or dipole), you can manually rotate them (indoors) and they can be very directional, good for mostly blocking that same-frequency station that may be coming in on top of the main one you want. There are some plans on the web.
regardless if ur going to build a dipole, a long wire, or a loop I'd consider getting a short wave radio also. Just so ya wouldn't be limited to the AM Broadcast Band. u'd get dual duty from both the s/w radio, and the antenna.
^ Yes, noise pickup is the problem with many AM antennas, especially bad in an urban environment (like where I am). I quite like the loops for this. Don't compare the crappy (but still directional) loops you get with AVRs etc. with the larger ones, there is a huge difference. I am speaking of indoor loops here, never tried an outdoor one (I don't know why...).
A big spool of something very fine (between 20AWG and 24AWG) will do great. Anything bigger is too cumbersome and doesn't help. If you have an old TV around you could carefully remove the yoke from the picture tube and use that great supply of fine copper wire. Old AM Radios have coil antennas with the perfect wire for your use.