(1) This gave powerful results such as a purely algebraic proof of the Riemann Roch theorem.(2) His work in algebraic number theory led him to study the quaternions and generalisations such as Clifford algebras.(3) In his lectures Sylow explained Abel's and Galois's work on algebraic equations.(4) The papers look at algebraic curves, the Riemann Roch theorem and algebraic polynomials.(5) Paniker not only played upon the abstraction of the words and scripts but also incorporated mathematical formulae, algebraic equations and diagrams of horoscopes.(6) Arabic mathematicians learned to manipulate polynomials, to solve certain algebraic equations, and more.(7) Fermat preferred the algebraic techniques that he used to such devastating effect in number theory.(8) Albanese's research involved examining curves on algebraic surfaces and the genus of an algebraic variety.(9) Omar Khayyam, known chiefly in Europe as a poet, combined trigonometry and approximation theory to solve algebraic equations using geometry.(10) Even his analytical work was guided by algebraic and linear algebraic methods.(11) To do this, he advanced the study of differential equations from numbers, formulas, and the manipulation of algebraic equations to geometry, curves, and the visualization of flows.(12) He also published results on algebras which were fundamental in the study of algebraic number fields.(13) The main innovation was Playfair's use of algebraic notation to abbreviate the proofs which he taught in his class.(14) Naimark's first work for his candidate's thesis was on the separation of roots of algebraic equations.(15) These books allowed new generations of mathematicians to learn algebraic topology and homological algebra directly in the categorical language and to master the method of diagrams.(16) algebraic function
