The French-born 21-year-old arrives at the Emirates Stadium after turning down a new deal with the Bundesliga side.
The Gunners have not disclosed any compensation fee to be paid to Bremen and could be forced to go to a UEFA tribunal to determine how much they owe the German outfit.
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Bischoff joined Bremen in 2005 after spending his early years in France with Colmar and Strasbourg.
However he made only one senior appearance for the German team, in a 2007 UEFA Cup tie, and made the decision to move on this summer.
The versatile midfielder will join up with the Gunners squad upon their return from an Austrian training camp.