If the goal of imperialism is to create an empire, why is the word not spelled “empirialism”?
Imperialism is first recorded in English in 1826, from imperial + ism.
The Online Etymology Dictionary says of imperial:
late 14c., “having a commanding quality,” from Old French imperial (12c.), from Latin imperialis “of the empire or emperor,” from imperium (see empire). Meaning “pertaining to an empire” (especially the Roman) is from late 14c.
And for empire:
early 14c., from Old French empire “rule, authority, kingdom, imperial rule,” from Latin imperium “rule, command,” from imperare “to command,” from im– “in” (see in– (2)) + parare “to order, prepare” (see pare).
So both English words have roots in the Latin imperium (at one stage or another) and come to English via Old French where they spelled one with an E and one with an I.