Algebraic solution of high degree polynomials In layman terms, why is it not possible to algebraically solve polynomials of degree five and higher?

One author phrased it thusly (more or less): In a polynomial, every root is an irrational function of the coefficients, but every coefficient is only a rational function of the roots.  This dichotomy sets up a sort of 'internal tension' between the roots and the coefficients, which increases with the degree of the polynomial.  By degree 5 the 'tension' has exceeded the representational capacity of nested root-extractions.
