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?



what part of using vieta's formulas violates quintic non-solvability?
The proper notion is "unsolvability with respect to a certain set of operations"; in the case of Galois-Abel's result regarding the quintic equation, this means that there will be no nice algebraic formula using just nth-roots, addition, etc. (Use Some encyclopedia for the proper set.) There are formulas for solving the quintic in terms of more advanced operations; again there are many easily found sources showing how and with what functions.


Similar problems that are not solvable at one level become solvable with more powerful tools, from squaring the circle to determining which Turing machines halt on a blank tape. You need to adjust your perspective to the situation and decide what is appropriate.


This type of situation arises often in mathematical logic, especially universal algebra. Look up reverse mathematics to see what you need for proving certain theorems; look up clone theory, interpretability theory, and classifying your favorite kinds of algebra for examples of what you can or can't do if given extra operations or relations.


Gerhard "Ask Me About System Design" Paseman, 2011.11.03


