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Gems

Gems go into item sockets and grant stats that depend on where they are socketed. You can swap them freely: removing a gem with Andvari is free, and it returns to your gem bag, separate from your item bag.

Gem Element In a weapon In armor In jewelry (amulet and rings)
Ruby Fire added Fire damage Fire Resistance % Fire Damage
Sapphire Ice added Ice damage Ice Resistance % Ice Damage
Topaz Lightning added Lightning damage Lightning Resistance % Lightning Damage
Emerald Poison added Poison damage Poison Resistance % Poison Damage
Amethyst Seiðr added Seiðr damage Seiðr Resistance % Seiðr Damage
Onyx Physical added Physical damage Armor % Physical Damage
Amber defensive Life per Hit All Resistance Dodge

Every gem comes in five qualities. Their values multiply those of the Chipped quality.

Quality Value Where it appears
Chipped ×1 from the start
Rough ×1.6 loot and combining
Cut ×2.4 loot and combining
Flawless ×3.4 from Ragnarök onward
Royal ×4.6 from tier 10 of the Expeditions onward

Combining: with Andvari, 3 identical gems become 1 of the next quality, for a gold cost beta:

Result Gold
Rough 100
Cut 400
Flawless 1,500
Royal 5,000
Royal Ruby
RubyFire
Royal Sapphire
SapphireIce
Royal Topaz
TopazLightning
Royal Emerald
EmeraldPoison
Royal Amethyst
AmethystSeiðr
Royal Onyx
OnyxPhysical
Royal Amber
Amberdefensive

The exact values for each gem and quality are in the Gem catalog. For sockets, see Andvari: Sockets and Gems.