Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on it.


So we know by Gaois theory that we can't write expressions for the solutions to all algebraic equations using the four arithmetic operations and nth roots. 


Question: is it possible to write expressions for the solutions to all algebraic equations using the four arithmetic operations and nth roots and something else in finite numbers of factors? For example adding logarithm to the set of operations? ( or other function, or class of functions...)? 


If Yes: Do they form any kind of algebraic structure?



method of finding roots of polynominal equations with arithmetic operations and roots and other functions
The answer is no: you need to keep adding more and more operations. For degree $n=5$ you can use elliptic functions (or the Jacobi $\Theta$ function, or Bring radicals - see below), for $n=6,7$ the Lauricella functions are needed (they are a 2-variable version of hypergeometric functions), and after that you need more and more complicated 'new' functions. You can apparently use the elliptic Siegel functions (aka Siegel modular forms) for the general case, but I've never looked into that. Most of this was all worked out in gruesome detail by analysts for decades, with lots of research on this up until the early 20th century (and then generally forgotten!).


For trinomial equations, you can use hypergeometrics to solve all of them. The derivation of that is fun.


It is hard to find information about this on the web. The 'best' discussion of related ideas is the Wikipedia page on the [Bring radical](http://en.wikipedia.org/wiki/Bring_radical). There is a good [timeline](http://library.wolfram.com/examples/quintic/timeline.html) at Wolfram's site.


