Galbatorix died being killed by Eragon in the 4th book, the Inheritance. He was stoke with a heart attack from Eragon's sword, and last he destroyed himself, attempting to take one of the team's member away with him, but this attempt failed since Eragon cast a spell before Galbatorix exploded himself out.
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In the fourth book in Christopher Paolini's Inheritance sequence, Galbatorix is cornered by Eragon and his allies in his palace. Eragon casts a desperate, non-worded spell designed to show Galbatorix all the pain he has caused, which drives him mad. The King then casts a simple spell that, translated, reads "Be not". This creates a nuclear explosion which annihilates him.
In the Inheritance Cycle book series by Christopher Paolini, Galbatorix is killed by Eragon using the name of the ancient language (the true name of magic). Eragon uses this power to sever Galbatorix's connection to the Eldunarí, causing him to lose control and implode.