The eastern half of the Roman Empire, initially ruled by Constantine I, was named the Byzantine Empire due to its largely Greek-speaking population. Before Constantine named the capital city Constantinople (330 AD), it was referred to as Byzantium. The other answers are incorrect because they refer to empires that either existed before (Persian), elsewhere during (Hunnic and Hephthalite), or after the Byzantine Empire's reign (Ottoman).

Valentinian I ruled the Western Roman Empire during the fourth century. He is sometimes remembered as the last good emperor, although this is most likely because the empire collapsed shortly after his reign. He, notably, divided the Roman Empire once again (it had been previously divided under Diocletian, before being somewhat reunified under Constantine) and placed his brother, Valens, on the throne of Constantinople.

The city of Byzantium (also called Constantinople) was made the capital of the Roman Empire in the fourth century by the Emperor Constantine. The city is located on the European side of the Bosphorus; a body of water which connects the Mediterranean and the Black Sea. It was thus perfectly situated to benefit from the trade between Europe and Asia. The city would grow immensely wealthy from control of this trade.

The Roman Empire had become too large to be ruled by one emperor by the third century (this was one of the causes of the Crisis of the Third Century). It was divided, by Emperor Diocletian, into a tetrarchy. This tetrarchy was then dissolved in favor of an Eastern and Western Roman Empire. During the reign of Theodosius I, in the late fourth century, the two empires were reunited once again under his rule. But, when he died, Theodosius left the eastern portion to one of his sons and the western portion to another of his sons. This was the final, permanent division of the Roman Empire into a western and an eastern half. The Western Roman Empire would not last another century, whereas the Eastern Roman Empire would continue for more than one thousand years.

The Roman Empire was divided into an eastern half and a western half in 285 CE by the Emperor Diocletian. It was the Emperor Constantine in 330 CE, however, who moved the capital of the Roman Empire to Byzantium (Constantinople), in the Eastern Roman Empire.

The Tetrarchy (“rule of four”) was created by Diocletian in 293 CE. In the third century, the Roman Empire had grown too large and impossible to administer. To solve this problem, Diocletian divided the Roman Empire into four distinct territories, each to be governed by a different ruler. This brought to an end the Crisis of the Third Century and stabilized the Roman Empire. The Tetrarchy did not last long, and within twenty years was dismantled, in favor of a permanent division between an Eastern Roman Empire (led by Constantine) and a Western Roman Empire (led by Licinius).

Diocletian was the fifty-first Roman Emperor and was one of the most significant. His leadership brought to an end the so-called Crisis of the Third Century. He notably created the Tetrarchy, dividing the Roman Emperor into distinct territories, each to be administered by a different ruler. He abdicated power in 305 CE, and due to competing claims for authority in the Roman Empire, a destructive civil war soon broke out.