No single person invented algebra. The roots of algebra extend to the ancient Babylonians, who first developed mathematical techniques to solve problems using algorithms. Later work by Greek, Persian and Indian mathematicians further extended these techniques and laid the foundations for modern algebra.
Two major figures in the development of modern algebra were Francois Viete and Rene Descartes. Viete created the first modern symbolic algebra that used letters to represent variables in equations. Descartes created the coordinate system that allowed geometric problems to be solved in algebraic terms. He also introduced the modern form of notation that is still used in algebra today.