The origins of algebra can be traced and seen as far back as the Babylonians. However, it was the Greeks, around the 3rd century, that started studying and expanding algebra at a more in depth level. Diophantus of Alexandria, the Alexandrian Greek mathematician, is often credited with being the first person to study and teach the field of algebra in a manner that resembles what we now know today. For this reason, he was given the nickname the “Father of Algebra”.
Diophantus wrote a series of books called Arithmetica which consisted of 130 algebraic problems and also featured detailed solutions for each. The equations throughout these books are often referred to as the Diophantine equations and they have served as the foundation for several modern day algebraic equations such as the quadratic equation.
After Diophantus, the next notable figure in the history of algebra is Persian mathematician and astronomer Muḥammad ibn Mūsā al-Khwārizmī. His mathematical text, originally written in Arabic around AD 820, was called Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala. The word algebra derives from the word “al-ğabr” in the title of this text. The full translation of the title is The Compendious Book on Calculation by Completion and Balancing and the text featured the first systematic solution of linear and quadratic equations. It is often debated that al-Khwārizmī deserves the title as the “Father of Algebra” but Diophantus is generally still given credit.
The whole of algebra can be broken down into the following more specific categories: elementary algebra, abstract algebra, linear algebra, universal algebra, algebraic number theory, and algebraic geometry. Elementary algebra is the basis for all of the more advanced forms and it is the first type of algebra taught in education to students with no more than basic math experience. Throughout elementary school, junior high, high school, and college, students are generally offered multiple different courses in algebra which get more advanced and in depth as they progress. You can find many great free math classes online.