Gaul wasn't a powerful empire.
Carthage was a settlement for retired soldiers.
Gaul
No, the capital of the Roman Empire was Rome. Gaul was a conquered territory, roughly corresponding to present-day France.
conquered Gaul (modern France) and set Rome on the road to Empire
From ancient Rome's early days and moving forward into the empire's expansion, one area seems to me to have been problematic for ancient Rome. In my opinion, Gaul (modern day France and the lowlands )was a problem. Yes were times when Gaul was the least of the empire's problems. The size of Gaul, however, and the richness of the land made Gaul not the easiest of areas for Rome to control. Other "uncivilized" nations liked being in Gaul, whether they were invited or not. Gaul had a large significant sea coast and its eastern borders were dominated at times by Germanic tribes. Because of Gaul's value to Rome, Gaul needed constant attention.
Caesar as a powerful general conquered France & Belgium, then called Gaul. Gaul was added to the Roman Empire. Julius Caesar also invaded Britain. Before his murder, he destroyed the army of his rival Pompey in Greece. Caesar while in total command of Rome, granted his allies in Spain & Gaul, Roman citizenship.
Alexandria would not be found on a map of ancient Rome. Alexandria was/is far to the east in Egypt. It would be found on a map of the entire Roman empire on the coast of Egypt, slightly to the west of the Nile delta.
Gaul not an Empire. It was not even a unified state. There were two Gauls: Translapine Gaul (France, Belgium Holland south of the River Rhine and Germany west of this river) and Cisalpine Gaul (in part of northern Italy.) The former was inhabited by some 54 independent major and minor tribes and the latter was inhabited by eight independent tribes. Alexandria was not an empire either. It was a city. At the time in question it was the capital of a kingdom: the Ptolemaic kingdom of Egypt. Carthage had an empire in the western Mediterranean: its homeland in Tunisia and western Libya, the islands in this part of the Mediterranean (Sicily, Sardinia, Corsica and the Balearics) and trading outposts in southern Spain. Following her first war against Rome she conquered southern Spain and the greater part of its east coast. Before the wars against Carthage Rome controlled Italy through a system of alliances with some of the Italian peoples and through founding settlements in strategic areas of the peninsula. She started developing an empire outside Italy by winning the three Punic Wars against Carthage, taking over her territories after each war.
The Gallic Empire (Spain, Gaul, and Britain) and Palmyra in Mesopotamia
Some people think that the Roman Empire was the most powerful empire in history. However, this is not the view of historians. More powerful empires have existed. Probably the most powerful empire in history was the British colonial empire.
The Persian empire was gone before the Roman empire became powerful.
The emperor in ancient Rome was an absolute ruler who was the most powerful man in the empire. At that time, the pope, intended as the Bishop of Rome, was not the most powerful clergyman. He was part of the Pentarchy. This was a term which conveyed the idea of universal rule over all Christians by the heads (or Patriarchs) of the five major episcopal sees of the Roman Empire: Rome, Constantinople, Alexandria, Antioch, and Jerusalem. These five patriarchs were equals.