There's a lot of ifs and depends that go into the answer. Let me clarify:
Will an i3 handle BF3? Depends. For single player, it'll do fine (like any modern CPU) for most multiplayer it'll do fine. For the big 64-player multiplayer maps the i3 will likely choke, and you'll be better off with a quad-core.
Will you be overclocking? Generally, you NEED to overclock AMD chips to get decent performance from them. This adds to cost (motherboard quality & aftermarket heatsink unless you like loud computing). So if you or your friend aren't comfortable with overclocking, then AMD is probably a bad idea.
Will other games be important to you that might affect your choice. I'm thinking specifically of StarCraft II, the Total War series of games, or Civ5, all of which benefit from Intel.
So your choices tend to come down to:
1) "I need things cheap, I don't really plan to do multiplayer on BF3 much, or at least not the really big maps, and yes performance in other games will matter to me"
clear win for the i3-2120
2) "I do play to do multiplayer BF3 on big maps, I can figure out an overclock no problem, and I want the best bang-for-my buck to make more room to spend $ on my graphics card, also I live near a microcenter"
clear win for the fx-4170 (or even fx-6xxx)
3) I do plan to do multiplayer BF3 on big maps, but what's this 'overclocking' thing? I/my friend are scared of changing things like CPU voltage settings.
Clear win for the unmentioned i5-3450, which you can pair with a cheap B75 mother board for a cpu+mobo cost of about $265. More expensive than the other options, but not by much considering overclocking a 4170 requires a good motherboard & an aftermarket cooler. This option means that any AMD option (including CPU cooler) you look at should total no more than $230 or so, or you're kind of getting ripped off, since an i5 will best anything AMD-related gaming.
Frankly, you can easily fit an i5-3450 into a $750 tower budget, unless you're also including things like a monitor or an SSD+expensive case, and it's probably the option you should go for.