Islamic civilization, the various golden ages of Islamic civilization always occur early in the first few centuries in which a new territory is occupied. Wherever the various Muslim vanguards invaded, the vast majority of the population was non-Muslim. It would take many years for this population to be converted and assimilated. These non-Muslims or recent converts are the ones who carried on the work which many historians are prone to attribute to "Islamic" civilization. Thus, a distinction must be drawn between the so-called high Islamic civilization and the religion of Islam. Eventually as the process of Islamization proceeds the non-Islamic component of the population becomes a small minority and stagnation sets in. This process is evident in the first centuries of the Arab conquests where the process of Arabization and conversion to Islam took a few centuries to complete; this was the "Arab" golden age, a product of unconverted or recently converted Christians, Jews and Zoroastrians.
Algebra had a more mixed origin; it was only partly derived from Hindu texts. The word algebra was also obtained from al-Khwarizmi’s book Al-jabr wa’l muqabalah. Moreover, in certain respects, the works of al-Khwarizmi were at a lower level than those of his Greek and Hindu predecessors. Boyer and Merzbach write: " ...in two respects the works of al-Khwarizmi represented a retrogression from that of Diophantus. First it is on a far more elementary level ... and second ... [it] is thoroughly rhetorical, with none of the syncopation found [in the works of Diophantus] ... or in Brahmagupta’s work. Even numbers were written out in words rather than symbols! … Nevertheless, the Al-Jabr comes closer to the elementary algebra of today than the works of Diophantus or Brahmagupta, for the book is not concerned with difficult problems in indeterminate analysis but with a straightforward and elementary exposition of the solution of equations, especially of second degree."
Thus, the Arabs must be credited not with inventing algebra, but with making it more accessible for the solution of simple problems. As for the ultimate origin of modern algebra there are three schools of thought: “one emphasizes Hindu influences, another stresses the Mesopotamian, or Syriac-Persian, tradition, and the third points to Greek inspiration. The truth is probably approached if we combine the three theories.” Historians of mathematics Boyer and Merzbach conclude: "It is probable that al-Khwarizmi typified the Arabic eclecticism that will so frequently be observed in other cases. His system of numeration most likely came from India, his systematic algebraic solution of equations may have been a development from Mesopotamia, and the logical geometric framework for his solutions palpably was derived from Greece."
The example of algebra is an ideal case illustrating the role of cultural cross fertilization in the short-lived period of high civilization under the early Pax Arabica. Algebra was derived from a combination of ideas developed by the oriental culture superseded by Islam, the classical learning of ancient Greece, and an impetus from a far-off land, in this instance India that became accessible due to the vast extent of the Arab empire. And, of course, it reached its full development in a land that still contained a majority population of non-Muslims and recent converts who were well versed in their ancient traditions.
Furthermore, the Hindus had a continuing role in the development of algebra subsequent to al-Khwarizmi as the civilization of the Arabs ossified under the deepening influence of Islam. The radical sign, and many algebraic symbols appear to have been invented by the Hindu mathematician Bhaskara in the twelfth century. For a comprehensive view of the sources of the so-called high Islamic civilizations see Islamic Expansion and Decline.