what was this period's contribution in the fields of science and medicine?
In the field of engineering Isidore of Miletus, the Greek mathematician and architect of the Hagia Sophia, produced the first compilation of Archimedes' works; Pendentive architecture, a specific spherical form in the upper corners to support a dome, is a Byzantine invention; A mechanical sundial device using the Antikythera mechanism was reactivated
Byzantine Empire
In the field of engineering Isidore of Miletus, the Greek mathematician and architect of the Hagia Sophia, produced the first compilation of Archimedes' works c. 530, and it is through this manuscript tradition, kept alive by the school of mathematics and engineering founded c. 850 during the "Byzantine Renaissance" by Leo the Mathematician, that such works are known today (see Archimedes Palimpsest).

Pendentive architecture, a specific spherical form in the upper corners to support a dome, is a Byzantine invention. Although the first experimentation was made in the 200s, it was in the 6th century in the Byzantine Empire that its potential was fully achieved.

A mechanical sundial device consisting of complex gears made by the Byzantines has been excavated which indicates that the Antikythera mechanism, a sort of analogue device used in astronomy and invented around the late second century BC, continued to be (re)active in the Byzantine period. 
which region was part of the eastern roman empire The extent fluctuated, but at its peak, it stretched from Byzantium up to Syria and Italy and North Africa, how would you define this type of empire? Romano-Hellenistic who are some of the rulers during this period? Justinian I, Constantine, Julian, Theodosius I to name a few what languages were spoken in the eastern region during this period? Primarily Greek, but Latin was introduced later
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Byzantine Empire [SEP] Science, medicine and law
The writings of Classical antiquity were cultivated and extended in Byzantium. Therefore, Byzantine science was in every period closely connected with ancient philosophy, and metaphysics. In the field of engineering Isidore of Miletus, the Greek mathematician and architect of the Hagia Sophia, produced the first compilation of Archimedes' works c. 530, and it is through this manuscript tradition, kept alive by the school of mathematics and engineering founded c. 850 during the "Byzantine Renaissance" by Leo the Mathematician, that such works are known today (see Archimedes Palimpsest). Pendentive architecture, a specific spherical form in the upper corners to support a dome, is a Byzantine invention.