Did I err in praying Dhuhr a little before the Adhan for Jumu'ah because of a conflict between prayer time and an exam, and must I make it up, and what should I have done?

The Dhuhr prayer is not valid before the first Adhan (definition) if the first Adhan occurs at the very beginning of its time or before it, because that would be praying before the appointed time. Exams are not an excuse to miss Jumu'ah unless there is significant harm. The Dhuhr prayer is not valid, according to many jurists, if performed after the time for it has begun but before the Imam has prayed Jumu'ah. Rather, one must repeat the Dhuhr prayer after the Imam has prayed Jumu'ah, and repent for missing Jumu'ah. However, if you have a valid excuse for missing Jumu'ah and you pray Dhuhr after its time has begun, then there is no blame upon you.