Peterson departs after catching the eye during pre-season.
The 19-year-old looked impressive in pre-season and scored in friendlies against Brondby and Preston North End.
His performances impressed so much that he was included in the squad's pre-season tour of the United States when he was initially overlooked.
It looks as though his opportunities will have been limited had he stayed at Liverpool, with the chances of any first team action perhaps being limited to League Cup ties if anything. His last appearance in a Liverpool shirt came in an under-21 game against Manchester United at Anfield - which finished 1-1.
The club has invested heavily this summer to have a big squad to compete on all four fronts. The first team currently has Daniel Sturridge, Mario Balotelli and Rickie Lambert at the club for sure for the season, whilst it looks increasingly likely that Fabio Borini will stay put despite being linked with, and having an offer accepted for a move to Sunderland this summer.
The club's youth side also has some promising attacking talent with the likes of Jack Dunn, whilst Samed Yesil is still playing with the under-21 side and has made a couple of appearances for the first team - though has subsequently started to recover from a serious knee injury.