Notice! You can attempt this quest only after completing the only main quest of The Bards College, namely Tending the Flames.
Head to The Bards College in Solitude and find Pantea Ateia (screen above). She will inform you that the title flute has been stolen and that her former student Larina is behind it. Pantea doesn't intend to buy the flute back and will instead want you to take it from a certain necromancer who is now in possession of it.
Head out of the College, open the world map and plan your journey to Hob's Fall Cave (screen above). It can be found in the northern part of Skyrim and in case you haven't been there yet, it would be best to start at Winterhold or Dawnstar. It's best to approach the cave from the south, choosing the path running between two large glaciers.
Head into the cave and start exploring its narrow corridors and little rooms. On your way you will be attacked by Skeletons and Necromancers (screen above). I'd especially suggest looking out for the Necromancers, as they're capable of not only casting offensive spells, but also summoning more undead or raising their fallen companions. In each of the fights, you should first eliminate the stronger necromancers and afterwards the apprentices.
Eventually you should reach a bigger room in the north-east part of the cave. Fight with the most numerous so far group of enemies, focusing mainly on attacking the very dangerous Ascendant Necromancer. Afterwards look around for a locked chest and head east. The passage here is blocked (screen above), but puling the chain on the right wall will solve this problem.
Look out for ice traps as you move towards the last room, they can be switched off by taking the soul gems from pedestals. After reaching the marked location, the last fight with Necromancers will start. I'd especially suggest focusing on attacking the Master Necromancer (screen above), as he can easily revive his apprentices. Keep an eye on your health level, as you can also easily die because of his strong ice spells.
Afterwards take a look around the area and note that Larina is already dead. Take Pantea's Flute from the big chest (screen above) and leave the cave after exploring it. Return to The Bards College to return the flute to Pantea. She won't have any material prize for you, BUT she will raise all six mage skills by 1 point (Alteration, Conjuration, Restoration, Enchanting, Destruction and Illusion.