The 24-year-old sealed the €50 million transfer on Monday afternoon having undergone a medical in Munich, where he is on international duty.
"I am thrilled to be joining a club of the stature of Arsenal and am looking forward to playing in the Premier League," he said.
"It will be great for my own personal development as a player and I am particularly looking forward to working with Arsene Wenger.
"From our negotiations it is clear the club has huge ambition and I look forward to being part of an exciting future."
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