Page "Fundamental theorem of algebra" Paragraph 9

A first attempt at proving the theorem was made by d ' Alembert in 1746, but his proof was incomplete.

Among other problems, it assumed implicitly a theorem ( now known as Puiseux's theorem ) which would not be proved until more than a century later, and furthermore the proof assumed the fundamental theorem of algebra.

Other attempts were made by Euler ( 1749 ), de Foncenex ( 1759 ), Lagrange ( 1772 ), and Laplace ( 1795 ).

These last four attempts assumed implicitly Girard's assertion ; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p ( z ).