It sounds like you're already SOMEWHERE with getting to know her... at least you both know you're inclined, so that's a step forward!
If she wants to get to know you, don't think she'll give up soon. But don't make her wait forever, either. Quite honestly, if she likes you and WANTS to get to know you, there are no rules. You can be random, it's ok. ask her bizarre questions (bizarre excluding ANYTHING sexual or pervy... but you knew that). in fact, if it's slightly out of the ordinary, it's better because it's exciting and new. that's how beginning of relationships are: exciting and new, and almost anything -as long as decent- is fresh and a good omen.
And keep in mind that she might be just as freaked out as you are (just hiding it better!). Her friends seem daunting... when she's in a group, go up to her and tap her shoulder (lightly!) and ask her a class related question as an excuse to speak with her. maybe her friends will keep walking forward, so then she'll be alone. if that's the case, ask her whether she's free that afternoon. you said she goes to the field often? maybe you can offer to go for coffee right after, just for an hour or so, casual. choose somewhere you're comfortable.
if they DON'T walk forward, maybe if YOU start walking away a little bit, she'll follow you and you can continue the conversation. or just quietly ask her to move away from them a bit so you can talk.
if her friends are jealous... that's ok! in fact, it's actually BETTER. you can bet your [butt] she's probably DOUBLY flattered by your attentions because of it! She got the guy, not them! So make it to your advantage: you have every right to like her, you shouldn't feel guilty for not choosing her friends, and she's worth every bit of your attention!
anyways, this is long... but if you ever feel like the awkwardness will never end, let me reassure you: it will. it's GOING to be uncomfortable in the beginning, but it's worth every moment that WON'T be uncomfortable. you've just got to make that first step and realise you don't look like a fool doing it. even if you stumble and go bright red in the face, you're NOT looking like a fool, you're making her feel better (because she realises she's not alone in being nervous) and more inclined to help you (by accepting your invitation!). besides, she'll probably be so nervous herself wondering if she's acting the right way that she won't even notice.