Short answers, pulled from the story.
The word algebra originates from a medical term meaning the setting of broken bones. It entered the English language in the 16th century through Italian and Spanish translations of the Arabic root meaning to restore or set right. The Persian mathematician Muhammad ibn Musa al-Khwarizmi published a treatise titled The Compendious Book on Calculation by Completion and Balancing in the 9th century which introduced the method.
Mathematicians began using letters to represent unknown quantities before the 16th and 17th centuries. The French mathematicians François Viète and René Descartes revolutionized the field by introducing letters and symbols to denote variables and operations. This symbolic formalism allowed for the manipulation of indefinite quantities and remains the standard today.
The Italian mathematician Paolo Ruffini and the Norwegian mathematician Niels Henrik Abel demonstrated that no general solution exists for polynomials of degree five and higher in the early 19th century. This impossibility theorem is known as the Abel-Ruffini theorem. Carl Friedrich Gauss proved the fundamental theorem of algebra at the end of the 18th century which described the existence of zeros for polynomials of any degree.
A group is defined as an algebraic structure with one operation that is associative and includes inverse elements for every member. Ring theory expanded this framework to include two operations that work similarly to addition and multiplication. The set of integers with addition forms a group where zero acts as the neutral element.
Linear algebra provides a method to interpret systems of equations as geometric figures where each equation represents a line in two-dimensional space or a plane in three-dimensional space. The solution to a system of linear equations is found at the point where these lines or planes intersect. This geometric interpretation extends to higher dimensions where equations with more variables correspond to higher-dimensional figures.
The collaborative effort to classify finite simple groups was mostly published between 1960 and 2004. This project took more than 10,000 journal pages to complete and stands as one of the most important mathematical achievements of the 20th century. The classification relies on the Feit-Thompson theorem which was a key early step in understanding the nature of these groups.