The archimage is obtained when a student has high mastery of at least 4 schools of magic.
Guide to Obtaining Archimage
Starting the Game
I advise you to launch a normal game in very easy, so that students are more focused on their studies and pick the option choose the magic books.
Then go to the lake region, because the special faction will give you big learning bonuses when you complete their quest.
Finaly select the magic books that upgrade the learning speed or reduces the needs for your students.
Building the School
Build your school keeping in mind what you picked when you laucnhed the game, does your arcanum classroom boosts the learning speed of nearby rooms? put it in the center of the major learning classes. Are your classes more efficient the higher they go? put them on the top of your school and let the dormitories on the first floor etc...
Don't forget that your refectories are more efficient next to a stable or the herborist classroom, so keep room next to them when you build them, then choose the right rule in the directors quarters.
When you create the houses you should keep an eye open for the houses with the bonuses that are effective with learning everything.
If you create an other house with funny people in it, it will help the other sutdents learn faster.
When you build learning classes always choose the teacher with the best learning skill even if you have to pay a bit more.
Everything should be done by now.
Now you just have to build all the classrooms needed and upgrade them.
You might want to buy an aquarium to start the quest with the special faction Nessie. If he came before you could buy one, just tell him to wait.
After you give him some fish, you will have to buy 4 dragons and put them in the school, they should be of different color. You can put them in the school before the aquarium. Nessie will then give you 40% faster learning for the rest of the game.
When you can go on the map, go to the merchants camp when you are high enough on gold (if not ask for the king when starting). At the camp buy teaching rooms to upgrade your school.
Since teaching rooms can be bought on the map and with the mana, buy stress releaving rooms or houses if you dont have a good one for the archimage when you buy cards with gold.
Pay your teachers maximum wage.
Put all your students in the House you build for the archimage, if you found funny sutdents put them in a House build for them, giving them bonuses when they talk to other students.
Now just wait for them to finish their learning course and you will have one.
You should do it in an hour or so.