Carn Dûm was the capital of Angmar.
Established on the very Eastern tip of the mountains of Angmar, on the Western tip of the Mountains of Angmar which is near the Dwarven holy site Mount Gundabad, it was inhabited by the Witch-king of Angmar and his many servants: fallen Men, Orcs, and other fell creatures. From here he based his attacks on the three petty realms of Arnor, quickly managing to destroy Rhudaur and later Cardolan.
Arthedain fell in the Battle of Fornost in Third Age 1974, but the Witch-king was routed by Glorfindel and in 1975 Angmar itself fell under an assault, and Carn Dûm was sieged by the Dúnedain of Gondor, the survivors of Arnor, and Elves of Lindon. Carn Dûm was utterly destroyed, although some Orcs survived there until at least the War of the Dwarves and Orcs and possibly after as well, until they were wiped out by Aragorn Elessar, High King of the Reunited Kingdom.
Carn dúm in Gaelic can be translated as 'mountain fortress' but it is not known if this was intentional by Tolkien since he didn't use Gaelic anywhere else.
In the context of the languages of Middle-earth, the meaning and the language of the name is not known. The circumflex ^ is used by Tolkien in Sindarin extra long vowels, but these words (if Sindarin) have not been attested.
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