You can write the n roots of an n degree polynomial in terms of its n coefficients, i.e., "Vieta's" formulas.

You can solve this system of nonlinear equations using Newton's method and the Jacobian.

What I am missing is which part of this procedure violates n>5 unsolvable algebraically --aren't all the matrix operations algebraic?

Is it taking the derivative that is the non-algebraic step that lets us solve?

In what sense is a derivative an "operation", what characteristics does it share with the other 5? Is there a group with nth derivative as its binary operation? Where does derivative fit in some kind of framework of algebraic operations? Quotient is an operation but is obtaining the remainder also an "operation" or would this also be in a separate part of the framework to the other 5? What are some references that address the placement of different operations in some kind of theoretical framework?