The implications of the insolvability of certain polynomials So I've just been over a bit of Galois theory, and I'm trying to understand what the implications are for a polynomial's Galois group to not be solvable.  My book says this means that there is no formula for finding the roots of the polynomial using just the field operations and the extraction of roots.
So does this mean that there is no one single formula, like the quadratic formula, which one can apply mindlessly to unsolvable polynomials in order to find their solutions?  Or does it mean the stronger statement that no matter how long you spend messing around with the poly using the field operations and the extraction of roots you will NEVER be able to figure out its solutions?
I've been under the impression that the roots of a polynomial with integer coefficients, that is to say algebraic numbers, could always be written in the form of some finite combination (sum/product/difference/quotient) of integers under roots of varying degree.  This is correct right?  Or are there algebraic numbers which must be written in terms of say
an infinite series?
And assuming there aren't such algebraic integers, how does one generally go about finding the roots of unsolvable polynomials with integer coefficients?  Just brute force and cleverness?
Thanks.
 A: If the polynomial $p(x)$ is irreducible over $K$ and has unsolvable Galois group over $K$, then no expression involving the coefficients, the elements of $K$, addition, multiplication, division, subtraction, and extraction of roots, will equal the roots of $p(x)$. 
In particular, there can be no general formula à la quadratic formula for the roots. 
Since there are monic polynomials with integer coefficients that have unsolvable Galois group, there are certainly algebraic numbers/integers that cannot be given as an expression involving rationals, addition, multiplication, division, subtraction, and extraction of roots. 
As to "finding roots of irreducible polynomials with integer coefficients", it really depends on what you mean by "finding". There are a lot of good methods to approximate roots (both real and complex) to any desired degree of accuracy (e.g., Newton's method). But generally, we don't write down 'expressions' that give them "exactly" in some sense.
(In a sense, what does it mean that we can "find" the roots of $x^2-2$? We write them out as $\sqrt{2}$ and $-\sqrt{2}$, but all we've done is give a name to those roots, without actually "exhibiting" them; we can compute $\sqrt{2}$ to any degree of accuracy we wish, but we can never write it down exactly, except by cheating and writing things like "$\sqrt{2}$").
A: One implication is that finding eigenvalues is a non-linear problem for which there is no finite algorithm, only numerical approximations. (Actually, finding eigenvalues and finding roots of polynomials are equivalent problems, because of the companion matrices.)
A: The stronger statement is true.
You are right to wonder about the possibility that every individual polynomial might have a root despite there not existing a general formula, but it turns out that there are, in fact, polynomials whose roots cannot be written in terms of field operations and taking roots.
There are lots of other ways of representing numbers than infinite series. One useful method for an representing a real algebraic number $\alpha$ is in the following way:
$\alpha$ is the unique root of the polynomial $f(x)$ lying in the interval $(a,b)$
Mathematica, for example, can do computations with a similar form.
A: There is no general formula for degree five and more, and even for the third degree there are so called irreductibiles cases with no rational formula with radicals.
When $|y|<1$,
$$4x^3-3x=y$$ is solved by $y=\cos\dfrac{\arccos y}3$, which is a transcendental function.
Anyway, that doesn't mean that the roots of polynomial equation of high degree are never expressible in closed form. Just think of $x^5-15x^4+85x^3-225x^2+274x-120$ $=(x-1)(x-2)(x-3)(x-4)(x-5)$.
The are numerous numerical methods for polynomial root finding and this is a broad topic. They are iterative. A related topic is root isolation, i.e. finding intervals where a single root is guaranteed.
