Solving algebraic equations that cannot be solved in radicals

[Demystifier]

In general, algebraic equations cannot be solved in radicals (we know it, e.g., from the Galois theory). So how can we solve such equations? We can always solve them approximately on a computer, but that's not what I'm asking about. Is there an exact analytic method to solve at least some of algebraic equations that cannot be solved in radicals?

 

[jedishrfu]

From searching I found this web page that lists some algebraic equations with links to exact solutions:

http://eqworld.ipmnet.ru/en/solutions/ae/ae-toc1.htm

and some exact solutons for systems of algebraic equations:

http://eqworld.ipmnet.ru/en/solutions/ae/ae-toc2.htm

 

[lurflurf]

In short you need more functions.
For example for the quintic
https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
https://en.wikipedia.org/wiki/Theta_function
https://en.wikipedia.org/wiki/Bring_radical
see
https://en.wikipedia.org/wiki/Quintic_function

 

[nuuskur]

With some guess work involved, one can factorise any polynomial as a product of first and second degree polynomials (over $\mathbb R$, that is).

Obviously, if $x^N-A$ has roots, then they are simple to find. In general it is known, that most of the higher degree polynomials don't admit any kind of parametrised solution.

See "Beyond the quartic equation" by R. King for some techniques for higher order polynomials involving modular functions. Applying this theory is, as one might expect, very difficult.

 

[Vanadium 50]

I'm not sure what sort of answer you're looking for. Obvioously one could define a function f(n, c1, c2...) to be the n-th root of a set of polynomial coefficients, but that's almost certainly not what you want. And we know radicals won't work.

 

[jedishrfu]

When I first learned DE, I recall the prof describing the various types of equations and the methods of attack. For algebraic ones, I expect the OP is interested in a similar cookbook so one can look at an equation and decide whether it can be solved exactly or a numerical solution is required,

Once in a Classical Mechanics class, there was a problem in the chapter on numerical solutions that posed a differential equation that was always solved numerically but one student found an exact solution which surprised the prof. Later it was found that the student had rediscovered an exact solution from a decades old CM book. Apparently, the equation lent itself to being solved numerically and hence the reason it was used in a chapter on numerical methods.

 

[Demystifier]

I'm not sure what sort of answer you're looking for.

I mean solved in terms of some functions which are not made up only for the purpose of solving these equations. For example, @nuuskur mentioned modular functions, which is the kind of answer I'm looking for.