At the end of this walkthrough, your character will own this Ancient Nordic Pickaxe, used to mine Stalhrim in the Skyrim Dragonborn Expansion:

The Ancient Nordic Pickaxe that is used to mine Stalhrim in Skyrim Dragonborn DLC

Talk to Glover Mallory, in the market area in Raven Rock, to begin the Ancient Nordic Pickaxe Quest. Glover Mallory is Raven Rock's Blacksmith. If he is not near the forge or otherwise missing, wait a few hours or travel elsewhere, then come back and check again.

Glover asks you if you have seen Crescius Caerellius, who apparently has taken Glover's pickaxe. Ask why all the fuss over a simple pickaxe.  Glover Mallory tells you that this isn't just your run-of-the-mill pickaxe... this is an Ancient Nordic Pickaxe. They don't exactly grow on trees, you know.

Ask Glover Mallory if he could just get the guard to find Crescius. Glover tells you he prefers to handle these things on his own. He tells you that if you see Crescius, tell him to give back the pickaxe. Glover will pay you for your trouble. You receive the miscellaneous quest: Retrieve the Ancient Nordic Pickaxe

Go to the Raven Rock Mines, which are located close to the docks (if you leave the docks towards the large stairway in front of you, turn left and go up the road to arrive at the entrance to the mine). Enter the Raven Rock mine. Crescius Carrelius will be in a room not far from the mine's entrance. Depending on whether you have already been in this mine or not, Crescius will either be arguing with his wife, Aphia Velothi, or sitting near his desk.

Talk to Crescius Caerellius. Choose the option: "Glover wants his Ancient Nordic Picaxe back". (If you do not have that option available, try completing the quest that Crescius has for you. Or you could try to pickpocket the Ancient Nordic PickAxe from him).

Crescius is not happy to hear Glover wants the axe back. He calls Glover a damn fool who does not even deserve to have it. The pickaxe was made for mining, not selling. Crescius thinks that Glover stole the pickaxe from the Skaal in the first place.

Your dialog options now will be:
It doesn't belong to you
Fine, you keep it
Let me think about this

Tell Crescius that the Ancient Nordic Pick Axe does not belong to him. Crescius turns over the Ancient Nordic Pickaxe to you, reluctantly. You should see the following messages now:

Ancient Nordic Pickaxe Added
Completed: Retrieve the Ancient Nordic Pickaxe
Return Ancient Nordic Pickaxe to Glover Mallory

NOTE: You probably know by now that the Ancient Nordic Pickaxe is used to mine Stalhrim, and are wondering whether or not to turn over the Ancient Nordic Pickaxe to Glover Mallory, or whether to keep it and use it for Stalhrim mining. But don't worry, Glover Mallory WILL LET YOU KEEP THE ANCIENT NORDIC PICKAXE, even if you turn in the quest. So go ahead and speak to Glover and tell him that you found his Ancient Nordic Pickaxe. If you still don't believe us, feel free to save the game before turning in the mission.

Return to Glover Mallory at the Forge in Raven Rock, tell him you have his Ancient Nordic Pickaxe. He tells you that after all the trouble you've been to tracking down Crescius, you might as well keep the Nordic Pickaxe to yourself. He hopes you will put the pickaxe to good use.

This completes the objective "Return Ancient Nordic Pickaxe to Glover Mallory". You now possess the Ancient Nordic Pickaxe, which you can use to mine Stalhrim!