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"?

  • $\begingroup$ This is a really cool question. Thanks for asking :) $\endgroup$ – Tyler Oct 8 '13 at 17:23

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.


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.


protected by Community Oct 11 '13 at 8:11

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