There are two different ways to become archmage in the mages guild. The first is to hit rank eight and have done all of Skink's quests, then talk to him and you'll get to fire the current archmage Trebonius. This is the peaceful method. Ignore it
The second is to do all of Edwinna's quests and be rank eight, then talk to Trebonius and he'll challenge you to a duel. This is absolutely the way you want to go. Why? 'Cause he has an enchanted amulet better then anything you can make, that you get to loot off his corpse.
The Necromancers Amulet
+25 to intelligence, constant effect
+25 resistance to normal weapons, constant effect
+25 spell absorption, constant effect
+1 restore health, constant effect
The cumultive enchantment values of the above are far, far higher (more then quadruple) what you would be able to put into enchanting a top-level amulet yourself. The spell absorption alone is worth more then 300 points.
He also has a decent staff with a huuuuuge (6500) number of charges, but rather weak enchantments.