Aside from the Mage's main story being a huge let down, finding out there is no Alchemy School in the College made me so damn sad. That was the whole reason I joined up.
I just assumed there would be Alchemy teacher and when you hit 100 you get to go do something special to get new recipes and (or) ingredients.
In Oblivion there was an Alchemy School but none in Skyrim’s Mage’s College.
I assume it's because alchemy is under the thief skill trees and not the mages which I find quite odd wasn't alchemy part of mage's skills for past elder scrolls games?
The Alchemist in Whiterun is a great one to "learn" from and youcan actually train from her. If you want some alchemy questing look to, the White Phial im Windhelm or the Alchemist's shop in Riften.