The inflaton field's lowest energy state may or may not be a zero energy state. This depends on the chosen potential energy density of the field. Prior to the expansion period, the inflaton field was at a higher energy state. Random quantum fluctuations triggered a phase transition whereby the inflaton field released its potential energy as matter and radiation as it settled to its lowest energy state. This action generated a repulsive force that drove the portion of the universe that is observable to us today to expand from approximately 10−50 metres in radius at 10−35 seconds to almost 1 metre in radius at 10−34 seconds.
The name inflaton is not missing an 'i' in the last syllable. It is spelled that way in order to follow the convention of field names, such as photon field and gluon field, which end with "on".