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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?

EDIT : My initial question reflected that "not really forgivable" confusion about the difference between algorithms and "analytic" process. These distinctions are interesting and not totally resolved at any level. In this case though the answer is pretty clear, but there are still interesting parts to be confused about around the edges.

Once that confusion was clarified, what seems really interesting to me is a framework in which to place the "bread and butter" of maths. Operations. I am interested in reading people who have already thought about this...I know there's a community standard to questions -- but what I really want to encourage on MO is that "interest and question asking" get rewarded. Not closed. Which is what happened here.

Seems like the establishment could benefit from and engage with questions from "the great outdoors". Or they could just occasionally deign to throw some scraps of inspiration to innocent questioners. Which is also kind of what happened here. But would have been better to not be closed. I would like to feel welcome on MO. But if I am not... then I guess it's back to reading some encyclopedia.....

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?

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?

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?

EDIT : My initial question reflected that "not really forgivable" confusion about the difference between algorithms and "analytic" process. These distinctions are interesting and not totally resolved at any level. In this case though the answer is pretty clear, but there are still interesting parts to be confused about around the edges.

Once that confusion was clarified, what seems really interesting to me is a framework in which to place the "bread and butter" of maths. Operations. I am interested in reading people who have already thought about this...I know there's a community standard to questions -- but what I really want to encourage on MO is that "interest and question asking" get rewarded. Not closed. Which is what happened here.

Seems like the establishment could benefit from and engage with questions from "the great outdoors". Or they could just occasionally deign to throw some scraps of inspiration to innocent questioners. Which is also kind of what happened here. But would have been better to not be closed. I would like to feel welcome on MO. But if I am not... then I guess it's back to reading some encyclopedia.....