How is the general solution for algebraic equations of degree five formulated?

Question

In one book one neural networks, I found the statement:

The general solution for algebraic equations of degree five, for example, cannot be formulated using only algebraic functions, yet this can be done if a more general class of functions is allowed as computational primitives.

What are the "more general class of functions"?

Answer 1

You can, for example, define an operation analogous to an $n$th root, except that instead of saying that $x=\sqrt[5] y$ if $x^5-y=0$, you say that $x=BR(y)$ if $x^5 +x -y=0$.

You can then express the solution of the general quintic in terms of $+, -,\times,\div,$ ordinary radicals, and $BR()$.

See Bring radical for more complete details, especially the section on solution of the general quintic.