The Reds received infielders Jose Peraza, Brandon Dixon and outfielder Scott Schebler from the Dodgers, with Frankie Montas, Micah Johnson and Trayce Thompson going from Chicago to L.A.
The White Sox recently signed infielder Brett Lawrie to a deal, and this move is expected to shift Lawrie from third to second base.
The 29-year-old Frazier recorded a .255 average with 35 homers and 89 RBIs in 2015, as well as winning the Home Run Derby at Great American Ballpark. The third baseman made the All-Star team in the National League each of the past two seasons.
Frazier expressed how excited he was to join Chicago, mentioning that he got really happy when he saw the projected lineup they will have on opening day. He can become a free agent after the 2017 season and has a salary of $7.5 million for next season.
Other players involved in the deal include Peraza, who was the Dodgers fourth-ranked prospect and 13th overall in the majors. The White Sox also sent out top-ranked prospects, including Montas ranked third, Johnson listed at fifth and Thompson ranked 14th.
Of all players not named Todd Frazier, Trayce Thompson saw the most playing time in the major leagues last season. He played in 44 games with the White Sox in 2015, hitting .295 with five homers and 16 RBIs.
By the end of last season, Frazier was the only one holding the Reds' offense up. Chicago also struggled offensively last season, but they are once again making moves in an effort to improve the numbers at the plate in 2016.