TITLE: method of finding roots of polynominal equations with arithmetic operations and roots and other functions
QUESTION [8 upvotes]: 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?

REPLY [4 votes]: Eisenstein in 1844 found a very simple hypergeometric series solution of the
quintic $x^5+x=a$, namely
$x=a-a^5+10\frac{a^9}{2!}-15\cdot 14 \frac{a^{13}}{3!}+20\cdot 19\cdot 18\frac{a^{17}}{4!}-\cdots$
Since it was shown by Bring in 1786 that the general quintic is reducible by
radicals to this form, Eisenstein's solution is probably the simplest solution
possible for the quintic.
For more on this story, see S.J. Patterson, Eisenstein and the quintic equation,
in Historia Mathematica 17 (1990), pp.132--140.