The promotion to Arch-Mage requires you to complete two quests – “Ambush” and “Confront the King”.
Ambush – Talk to Traven to receive this quest. He’ll send you to Silorn, southwest of the Imperial City, and meet the three Battlemages there. They will set up an ambush outside of the door to the ruins. Once you’ve positioned the Battlemages to your liking, speak to Thalfin and four necromancers will emerge from the ruins and attack you. During the attack, Falcar will flee back to the ruins. Follow him and chase him through the ruins until you can kill him. Take the Colossal Black Soul Gem off of his body and return to Traven.
Confront the King – Talk to Traven and he will promote you to Arch-Mage. He then kills himself and traps his soul in the Colossal Black Soul Gem. Take the gem off of his body, rendering you immune to the King of Worms spells that will turn you into a Worm Thrall. Travel to Echo Cave, west of Bruma, and kill the necromancer guarding the entrance. Fight your way through the cave until you find Mannimarco. He will trap you in a bone cage and talk with you for a bit until he tries to turn you into a Worm Thrall. When that fails, he will attack you. Kill him however you want, but be aware that he will frequently summon minions to aid him. Once he has been destroyed, return to Raminus at the Arcane University to finish the questline.