I am creating a Dragonborn warlock character and was just wondering if I could get a crystal arcane focus set into the middle of a gauntlet?
A magic item might be required for this
An arcane focus is usually a dedicated item.
An arcane focus is a special item [..] designed to channel the power of arcane spells. (PHB 151)
It is not intended for it to pull double duty (as armor or weapon). This setup would also allow for you to hold something in your hand without dropping the focus. So due to this, and possibly depending on what you want to use the gauntlet for, it could be reasonable for your DM to disallow this merger.
But there is a magic item that will help you if you intend the gauntlet to be a weapon, the Ruby of the War Mage (XGtE 138). It is exactly as you describe, a crystal you set into a weapon to make it function as an arcane focus. The downside is that it requires attunement.
Officially, no, but if handled correctly, it isn't broken.
The main issue with arcane foci is that they require the use of a hand; it's an object you have to hold and manipulate. This is one of the specific benefits of divine foci; they can be inscribed on a shield and let that item pull double duty. It would be giving you a benefit for free to allow you to use your arcane focus without restricting your ability to hold a shield, torch, or second weapon. Officially speaking, you can't do this.
That said, as a DM I'm always on the side of the player doing something cool, and a gauntlet with a gem in it is a really cool arcane focus. I'd probably allow this with the caveat that your hand has to be otherwise free in order to use the gauntlet as a focus. This of course comes with the footnote that I reserve the right to reverse my ruling if it later turns out to be somehow problematic -- perhaps because you've found some combination of feats or class abilities that takes this idea from "just a cool description" to "shenanigans".
Probably Not, but ask your DM
The whole point of components is to stop characters doing everything all at once.
Want to hit someone? You need a hand free to hold a weapon. Want to defend yourself? You need a hand free to wield a shield. Want to cast a spell? You need a hand free to make gestures and manipulate materials.
Want to do all three? Well, unless you are a thri-keen, you can't. Pick two.
Additionally, things held in a hand can be disarmed or stolen. This is a balance on the power of casters and fighters - the items they need (swords, axes, wands, orbs) can potentially be taken off them in combat. Allowing a component to be worn alters this balance.
On the other hand, disrupting this balance doesn't cause massive issues, especially if the same rules apply to NPCs, so your DM might say yes.
Ask them. After all, if you don't ask them the answer is definately "no".
RAW you can't, but talk with your DM for house rules.
Mechanically speaking, inserting a crystal arcane focus inside a gauntlet makes the gauntlet itself the arcane focus.
The description of arcane focus is in PHB (pag 151):
An arcane focus is a special item— an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item— designed to channel the power of arcane spells.
The description is quite general, and it does not include weapons or armors: by RAW a gauntlet can not be an arcane focus. One solution is in Szega answer: you may use the Ruby of the War Mage, but I don't think that applies to a gauntlet. The description says that you must insert it in a weapon and a gauntlet is usually part of an armor. In this case, ask your DM how to transform a gauntlet into a weapon.
Looking at the description of an arcane focus, you must hold it in one hand to use it: so gauntlet as an arcane focus allows you to use your hand (holding a weapon, a wand...), leading thus to unbalanced situations.
Nonetheless, there is some room for interpretation in the above description: one house rule could be that you can a gauntlet as an arcane focus provided that
- you must hold your fingers in particular position ( crossing them or closing your fist, for example) to activate the power of the gauntlet;
- instead of being allowed to always wear the gauntlet, you have to put it on before casting a spell and dorn it off after you finished your magic.
The second rule counterbalances the fact that this particular item can not be dropped (you can not be forced to drop it too) as may happen with an orb, for example. Even with these rules this usage looks still quite unbalanced: you should talk with your DM, if you both agree on this home rule you can give it a shot and see how it works, checking if it is gamebreaking. If so, remember that you can return to RAW and select a more traditional arcane focus.
Personally speaking, I think using a different item from the list provided in PHB as arcane focus is quite cool, if you can build a good background story or provide a nice motivation about why you are using it.