Mathematicians who contributed to Algebra?

Answer 1

Most recently, Ngô Báo Châ for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebra-geometric methods, for which he won a Fields Medal.

Also, from a historical perspective, and Egyptian named Ahmes produced a manuscript called the 'Rhind Papyrus' which showed the first simple algebraic methods.

There are many lists of mathematicians which can be found using a simple internet search.