The 26-year-old England international's contract at Anfield was drawing to an end but Madrid paid a fee to bring him in earlier so he can play in the Club World Cup.
Right-back Alexander-Arnold, who has just won the Premier League title with Liverpool, came through the academy of his boyhood club and won the Champions League in 2019.
He also won the Premier League in 2020.
The defender joins former Liverpool midfielder Xabi Alonso at Real Madrid, with the Spaniard appointed as their new coach to replace Carlo Ancelotti.
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