TITLE: How is the general solution for algebraic equations of degree five formulated?
QUESTION [22 upvotes]: In a book on 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"?

REPLY [10 votes]: Felix Klein has a small book called "Lectures on the icosahedron and the solution of equations of the fifth degree", where he develops a method of solving the quintic using modular forms. The relationship between the isocahedron and the general quintic is that the automorphism group of the isocahedron is $S_5$, which is also the Galois group of the general quintic.
Googling has turned up this introductory blog post on the topic, and this expository article.