Early arguments over it were faulty because they did not use complex numbers or, in Leibniz's case, did not realize that the square root of i could be expressed in the form a + b i, and thus failing in the case of x to the fourth power plus one.
Proofs of this result sometimes make use of Liouville's Theorem that a bounded entire complex function must be constant. An early proof of this by Argand was based on earlier work by D'Alembert, while the most famous proof of it was actually faulty because it assumed the Jordan curve theorem. Early arguments over it were faulty because they did not use complex numbers or, in Leibniz's case, did not realize that the square root of i could be expressed in the form a + b i, and thus failing in the case of x to the fourth power plus one. Nearly proven in Gauss's dissertation, name this theorem that states that every polynomial of degree n with complex coefficients has n complex roots.
Proofs of this result sometimes make use of Liouville's Theorem that a bounded entire complex function must be constant.
{Fundamental} Theorem of {Algebra} [prompt on partial answer]