How Isaac Newton Discovered the Binomial Power Series (2022)

> The method of high-school algebra is powerful, beautiful and accessible. So let us not be overwhelmed by the groups-ring-fields or the functorial arrows of the other two algebras and thereby lose sight of the power of the explicit algorithmic processes given to us by Newton, Tschirnhausen, Kronecker, and Sylvester.

and goes on to write:

> Personal Experience 1. In my Harvard dissertation (1956, [2]) I proved resolution of singularities of algebraic surfaces in nonzero characteristic. There I used a mixture of high-school and college algebra. After ten years, I understood the Binomial Theorem a little better and thereby learned how to replace some of the college algebra by high-school algebra; that enabled me to prove resolution for arithmetical surfaces (1965, [4]). Then replacing some more college algebra by high-school algebra enabled me to prove resolution for three-dimensional algebraic varieties in nonzero characteristic (1966, [5]). But still some college algebra has remained.

> I am convinced that if one can decipher the mysteries of the Binomial Theorem and learn how to replace the remaining college algebra by high-school algebra, then one should be able to do the general resolution problem. Indeed, I could almost see a ray of light at the end of the tunnel. But this process of unlearning college algebra left me a bit exhausted; so I quit!