Saarthal can be reached by heading into one mountain trails branching from the main roads leading to Winterhold. The player must speak to Tolfdir just outside the ruin's large entrance structure before s/he enters Saarthal proper with him for the purposes of this particular quest; note that the player does not need to wait for Onmund and J'zargo to also arrive, they will join the rest of the group automatically once the player is inside Saarthal.
Eventually Tolfdir will ask the player to speak with Arniel Gane; this College researcher will request the player to look for and find four items of note. Three are simply enchanted rings lying about on the ground, marked on the ground by small shining lights. The final item is an enchanted amulet still attached the wall; taking the amulet traps the player behind a gate, and short scene with Tolfdir follows.
With the amulet (now a quest item called the Amulet of Saarthal) equipped, do as Tolfdir instructs and cast a spell at area of the nearby wall that appears to be reacting to magical forces. This will break the wall, and lower the gate, allowing Tolfdir to accompany you as you both venture deeper into Saarthal via the new entrance you just opened.
Shortly after another scene where the player is introduced to a monk of the Psijic Order, Nerien, you will be attacked by Draugr. Fortunately, the player will have the assistance of Tolfdir as s/he explores the next portion of Saarthal. Once s/he has cleared a large chamber of Draugr, the player will need to head even deeper into the ruin alone while Tolfdir attempts to find out more about it.
After battling through more Draugr, the player will eventually encounter a lever and four pillars with engravings; naturally the player must set the pillars to show the correct engravings before the lever will successfully the open the gate leading beyond. Hint: look at the walls behind the pillars themselves for get the rather blatant answers. Clearly this was more a test of paying attention to your surroundings!
Another puzzle further on will require the player to activate the pillars in the right sequence in order have the proper engravings show on all of them; the challenge here is that changing some pillars also affects a number of others, potentially ruining any progress already made to getting the correct engravings to show. Patience and trial and error help here, but the methodical approach is to work backwards from pillars that affect the largest number of the others.
Tolfdir will eventually rejoin the player ahead of the final chamber of Saarthal, where they will discover the Eye of Magnus, and its Draugr "guardian" named Jyrik Gauldurson.
Jyrik is at first impervious to any form of attack at the beginning of the player character's battle with him. The key here is to last long enough for Tolfdir to discover what exactly is making Jyrik invincible; Tolfdir will then remove this barrier, allowing the player to finally defeat Jyrik.
Tolfdir will then ask the player to return to the College to inform the Archmage of what was uncovered at Saarthal.
A Word Wall containing a word for the Ice Form shout is located just before the exit, so be sure to acquire that as well.
 The Gauldurson Legend
Be sure to loot Jyrik's Staff and amulet fragment, as well as the Writ of Sealing. The latter two items are especially important because they are related to another quest, The Forbidden Legend. This side quest (unrelated to the College questline itself) will trigger at this point if it has not already done so.
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