The fifth main quest of the second act featured in The Elder Scrolls V: Skyrim is called Elder Knowledge.
The quest starts during your conversation with Paarthurnax, the dragon. You found that you need an Elder Scroll in order to kill Alduin, but you have to find its exact location.
The Elder Knowledge quest is longer than the previous one (The Throat of the World) and there are many locations you have to visit, including Alftand, Alftand Animonculory, Alftand Cathedral, Alftand Glacial Ruins, Alftand Ruined Tower, Blackreach, College of Winterhold, Hall of the Elements, The Arcanaeum, High Hrothgar, High Hrothgar Courtyard, Septimus Signuss Outpost, Sky Haven Temple, Throat of the World, Tower of Mzark and the Oculory.
During this quest there are also several adjacent objectives that must be completed, objectives that allow you to finish some side-quests such as the Word Wall Revelations faction quest, the College of Winterhold quest called First Lessons or the Discerning the Transmundane daedric quest.
As soon as you are ready to start the Elder Knowledge quest, you should ask Paarthurnax to offer additional info regarding the Elder Scroll. You will receive your first objective.
Objective 1: Learn the location of the Elder Scroll
Paarthurnax tells you that as soon as you find the Elder Scroll you will be able to meet Hakon, Gormlaith, and Felldir, who commanded the war against Alduin.
Now it is time for you to leave and find the Scroll. At this point you have two options:
Secondary Objective 1: Talk to Esbern
Secondary Objective 2: Talk to Arngeir
NOTE: If you already finished the Daedric Quest called Discerning the Transmundane, you can skip this main quest because you already have the Elder Scroll. In this case, your next main quest is Alduins Bane. Otherwise, no matter what secondary objective you decide to complete, you will find clues on the Scrolls location.
If you decide to visit Esbern travel to Sky Haven Temple and talk to him. He will tell you that you should visit College of Winterhold.
If you want to find the clues from Arngeir, travel to High Hrothgar and find him. He will also inform you that the mages of Winterhold may be able to help you, and that you should pay them a visit.
NOTE: If you continue your dialogue with Arngeir, you will be able to start a side-quest called Word Wall Revelations, as part of the Various Factions Quests set (The Greybeards).
Your next task is to travel to Winterhold; however, when you approach, you will see a bridge in front of you.
When you cross the bridge you will be stopped by a High Elf wizard, named Faralda. She will stop you, and tell you that its not safe to enter the College. You should avoid any hostilities while talking to Faralda.
Eventually, Faralda asks you why do you want to enter the city. Pick any answer you wish and, you will be asked to prove your magic skills.
If you are a member of the College of Winterhold, you can skip the test. Otherwise, you should complete it.
Persuade Faralda and tell her that youll be successful, and then agree to take the test.
When Faralda asks you to aim a spell at the seal on the ground, you have two options.
You can open the inventory and chose the spell requested by Faralda, then cast it but make sure you aim at the seal.
She will ask you to cast one of the following spells: Firebolt, Magelight, Fury, Conjure Flame Atronach, or Healing Hands, depending on the spells you know.
As soon as you complete this test, Faralda allows you to enter and tells you to look for Mirabelle Ervine inside the College.
You can ask her if you are allowed to enter as a Dragonborn. In this case, you will have to show her that you know the voice, and you must execute a Shout.
A good idea is to use the Fire Breath and aim below the seal. After you cast it, you will be able to enter the College and look for Mirabelle Ervine.
NOTE: In both cases, make sure you dont use a spell or a shout that has a large AoE (area of effect) otherwise you will attack Faralda and you will fail your quest. When you complete the secondary objective, you will start the College of Winterhold Quest called First Lessons, but keep in mind that this is a side-quest. It has nothing to do with the main quest and you can skip it if you want.
Enter the courtyard, open the large doors in front of you and then enter the Hall of the Elements.
Go right and you will enter the Arcanaeum.
Look around and locate an orc mage, Urag gro-Shub.
Talk to him and ask him about the Elder Scroll. He will be offended, and then he will give you a short briefing about the Elder Scrolls.
Ask Urag if there is an Elder Scroll you can use. He will start laughing at you then he will give you two tomes: Effects of the Elder Scrolls and Ruminations on the Elder Scrolls.
Read both books and you will start the Daedric Quest called: Discerning the Transmundane.
Objective 2: Ask Urag about the insane book
Return to Urag and tell him that the writer (Septimus Signus) of the Ruminations on the Elder Scrolls is crazy and you cant understand a word from his book.
He will confirm that Septimus Signus is indeed gone for a long while, but maybe he could help you if you find him in the Ice Fields.
Objective 3: Find Septimus Signus
Exit the College and go north, until you see a large iceberg. Make sure you jump across the floating ice and kill all Horkers, wolves and Ice Wolves that may attack you.
Septimus lives in a cave cut into the iceberg. Enter the cave by climbing down the ladder and you will see him standing in front of a huge box.
Ask him about the Elder Scroll and he will start a never-ending conversation with you.
Ask him where the Scroll is twice, and he will agree to tell you if you go to Blackreach (a Dwemer city located under Alftand).
Tell him that you agree and you will receive two items: the Attunement Sphere, and the Blank Lexicon.
NOTE: At this point, you will receive two objectives. To complete the first objective, you have to Transcribe the Lexicon (as part of the Daedric Quest: Discerning the Transmundane). The second objective is part of the main-quest you play; and to complete it, you have to find the Elder Scroll. Without the items from Septimus, you wont be able to complete the quest.
Objective 4: Recover the Elder Scroll
To reach Alftand, you have to travel to the mountains southwest of Winterhold.
During your journey, you will be attacked especially by wild animals.
The Alftand Ruined Tower is below the glaciers, and to reach it you must get to the Alftand Glacial Ruins.
The safest way to reach that point is to follow the wooden bridges and the platforms around the rooftops.
When you reach the entrance to the ruins, follow the tunnels down until you see a stone table. On the table, you will find several Research Notes and to the north, you will see a blocked door.
According to the notes, behind the door, may be someone.
Follow the passage west, and go up on the ramp, then down.
Kill the spiders and the Khajiit that attack you, and then proceed.
Inside the next room, you will see a cog and a piston and you will be attacked by two Dwarven Spheres.
Move north and enter the vent chamber and picklock the locked door, then go through the golden gate.
Follow the next passage and you will see another gold door, which is locked (Apprentice). Open it and pick up everything you find, then go around to the east, until you reach again the room with the cog and piston, but this time you are above its center.
Jump counterclockwise over the pistons and enter the corridor that takes you to the Alftand Animonculory.
NOTE: If you picked up too many Dwemer items, you should know that there is a wizard in Markarths Understone Keep, who pays a good price for them. His name is Calcelmo.
Continue through the tunnel but be careful to avoid the Dwarven Spheres.
Proceed through the Animonculory until you reach the Falmer nest.
A Falmer is a blindness creature and the degenerate remnant of the original Skyrim Elven inhabitants.
They will attack you as soon as you enter their nest, thats why you should sneak; however make sure you dont step on the bones you see on the ground.
Go down and you will enter a boiler room where you will see another nest.
Continue forward and down until you see a gold door on the southeast wall.
Follow the path down and kill the Falmer and the Frostbite Spider. Move through the next corridor and you will reach the Alftand Cathedral, but make sure you avoid the trap on the floor.
When you enter the cathedrals main chamber, clear it and then pull the lever above the entrance you used to get here. Stay alert because you will be attacked by a Dwarven Centurion. Kill it and get the key he is carrying, then reach the gate to the south-west, open it and listen to two thieves arguing: Sulla and Umana.
Slay the thieves and approach the strange mechanism.
Use the Attunement Sphere you got from Septimus, and you will see a hidden corridor. Follow the stairs down, and advance through the next gold door to enter Blackreach.
Follow the path west and pass the ancient structures and the stone bridge.
Turn left and you will see a large elevator that takes you to the Tower of Mzark.
Cross the corridor and open the gold door that allows you to enter a large hall, featuring a large sphere (Oculory). This device hides the Elder Scroll, but to get it you have to solve a puzzle.
Approach the platform across the Oculory and you will see a console featuring five cylinders: one is a Lexicon Receptacle and four buttons.
To solve the puzzle and get the Elder Scroll you must follow these steps:
Activate the Lexicon Receptacle until the Blank Lexicon is set on top.
To the Receptacles right look for the tallest pedestal and press it multiple times until the pedestal to the left of the middle one, starts to glow.
Look to the two pedestals to the left of the Receptacle and press the tallest one, twice, or until the button on the smallest pedestal starts to glow.
Now press the button on the smallest pedestal and watch the mechanism, which reveals a small container.
Jump down, approach the container and take the Elder Scroll, then exit through the door near the Lexicon Receptacle.
As soon as you pick up the Elder Scroll, the final quest of the second Skyrim act starts. The quest’s name is Alduins Bane.