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?