The quest is first triggered by speaking with Faralda when she is stationed at the gate leading into The College of Winterhold; she will prevent the player from entering and ask for their purpose, which should very likely be admission into the College! She will also quiz the player on what s/he intends to learn from the College, but the answer to this question will not affect your ability to actually join in any way.
To be accepted into the College, Mirabelle will then ask that you cast a spell from one of the five schools of magic at the emblem on the floor near where she stands guard. If you have not learned the spell she requests, she will offer to teach it to you for a firesale price of just 30 gold.
- Healing Hands (note that this spell needs to be cast on Faralda instead)
- Summon Flame Atronach
The required spell will be randomized in each new game, but you can change the spell to one you desire simply by saving your game before actually asking about the requirement, completely quitting and exiting the game, and then reloading the game (and your file). Considering the cost to learn some of these spells through purchasing the required spell tomes, there is a significant advantage in having Mirabelle teach it to you for just 30 gold instead.
No matter how you choose to acquire the necessary spell, once you have learned it, simply cast it as instructed; a trivial task compared to the undertaking to reach the College in the first place if you chose not to take any carriages to Winterhold itself!
Alternatively, the player can attempt to Persuade Faralda into gaining admission without actually undergoing the test, with success based on the Speechcraft skill level. Perhaps the easiest way to avoid having to take the test at all is to wait until the quest Elder Knowledge is triggered (as part of the Main Quest), in which case the player will gain entrance after proving to Faralda that s/he is Dragonborn by performing a Shout.
Upon successfully convincing Faralda, she will congratulate the player and welcome him/her as a new apprentice to the College. She will then ask that s/he follow her up the path while she lights magical beacons along the way. Upon reaching the College itself, Faralda will then ask that the player speak to Master Wizard Mirabelle Ervine as the next portion of his/her orientation.
After witnessing a brief conversation between Mirabelle and a Thalmor advisor, Ancano, she will present the player with an apprentice-level "uniform" (including a Novice Hood and a Novice Robes of Destruction) and will show him/her to her somewhat private quarters in the Hall of Attainment. The player will then be asked to enter the Hall of the Elements (the College's main building) to speak with Tolfdir.
Upon entering the Hall and the main chamber (past the iron gate), Tolfdir will greet the player and invite him/her to participate in an introductory lesson with him. The player also get the chance to meet the other three apprentices who will have entered the Hall just before the him/her.
The lesson serves as brief tutorial on casting Ward spells from the Restoration school of magic (a curious choice for Tolfdir, given he is a Master trainer of Alteration); the player will learn the Lesser Ward spell from Tolfdir if s/he has not already gained it through a spell tome. Move to the location as instructed by Tolfdir and cast the spell when ready; note that as a concentration spell, the spell must be kept active by holding the appropriate action key down until Tolfdir casts a Firebolt spell at him/her, completing the lesson and ending the quest.
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