After solving the Arkngthamz doors puzzle, two of the gates open.

The following objective is completed: Solve the Tonal Lock
The following objective is received: Claim the treasure of Arkngthamz
Go through either of the unlocked doors, and take the Aetherium Shard from the pedestal. Speak with Katria. She points out the edge has been cut, precisely cut. If you had another piece, about the same size, it would snap right in. We have to find the other pieces. There should be three more, one for each of the four cities that worked on the forge. Then, we still have to find the Forge itself, but Katria had a lead on that too.

Katria said she had a map in her journal, and suggested that's where you should start, so open Katria's Journal for the clues.

Your next objective is:
Search for the Aetherium Shards (1/4)

The first one you have already found after solving the door puzzle, and three remain.

Before leaving this chamber, note that there is a chest just behind the stand where you found the first Aetherium Shard, so be sure to loot it as it has pretty good quality treasure. Then, exit Arkngthamz the way you came in.

Check your quest log:

Lost to the Ages
In Arkgnthamz, I met Katria, a ghost searching for the Mythical Aetherium Forge. Together, we found one of the four Aetherium Shards that make up the key to the forge. She left search for the other shards and the Forge itself, and urged me to do the same.

As long as you set the quest to trackable in your journal, your map should point you to the correct locations. But sometimes the marker is bugged or does not appear. So see exact locations below:

Where to find the Aetherium Shards

Aetherium Shard locations are:

1st Aetherium Shard Location: Arkngthamz Puzzle Room (you should already have this)
2nd Aetherium Shard Location: Deep Folk Crossing (north east of Markarth)
3rd Aetherium Shard Location: Raldbthar (southwest of Windhelm)
4th Aetherium Shard Location: Mzulft (south east of Windhelm)

Note: You could get these shards in a different order, but unless specifically instructed to do so by your quest markers, we suggest you follow the order above to avoid the chance of glitches and bugs in the quest.

When you get to the location of the last shard, Mzulft, locate the Dwarven Storeroom building outside. Katria is waiting for you inside. The door in front of you will have an expert lock, but you can instead pick lock the door to the side which has an easier, apprentice, lock. Grab the fourth Aetherium Shard from its stand.

Once you have retrieved all four Aetherium Shards, the objective completes. In the next part of the walkthrough, you will be crafting your Aetherial Item!