Final answer:
When a trigger changes a field with history tracking, the history is updated with the change, even if the user doesn't have direct edit permissions. The update appears in the field history under the context of the trigger's execution, not necessarily the initiating user.
Step-by-step explanation:
If the current user does not have permission to directly edit an object or field, but activates a trigger that changes an object or field with history tracking enabled, the field history is still recorded. The change made by the trigger will show up in the field history as modified by whatever context the trigger was executed with, often the system or the context user that last updated the record before the trigger was called. However, it is important to note that the person or process triggering the update may not be the same as the user context under which the trigger operates.