Hellenistic mathematician Euclid details geometrical algebra in Elements.The origins of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algebraic fashion. With the use of this system they were able to apply formulae and calculate solutions for unknown values for a class of problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, and most Indian, Greek and Chinese mathematicians in the first millennium BC, usually solved such equations by geometric methods, such as those described in the Moscow and Rhind Mathematical Papyri, Sulba Sutras, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations.
Indian mathematicians proceeded to write treatises on algebraic means of solving equations from the end of the first millennium BC, followed by Hellenistic mathematicians from the early first millennium AD. Important algebraic works from this general era include the Bakhshali Manuscript, the works of Hero of Alexandria, the Arithmetica of Diophantus, the Aryabhatiya of Aryabhata, and the Brahma Sputa Siddhanta of Brahmagupta.
The word "algebra" is named after the Arabic word "al-jabr" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the Persian Muslim mathematician Muḥammad ibn Mūsā al-Ḵwārizmī in 820. The word al-jabr means "reunion". Al-Khwarizmi is often considered the "father of algebra" (though that title is also given to Diophantus), as much of his works on reduction are still in use today. Another Persian mathematician Omar Khayyam developed algebraic geometry and found the general geometric solution of the cubic equation. The Indian mathematicians Mahavira and Bhaskara, and the Chinese mathematician Zhu Shijie, solved various cubic, quartic, quintic and higher-order polynomial equations.
Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues.