Limitations to algorithm solvability: Galois methods and models of computation

We use simple arguments from Galois theory to prove the impossibility of exact algorithms for problems under various models of computation. In particular we show that there exist applied computational problems for which there are no closed from solutions over models such as Q(+, -, *, /, √), Q(+, -, *, /, k√), and Q(+, -, *, /, k√, q(x)), where Q is the field of rationals and q(x)ε Q[x] are polynomials with non-solvable Galois groups.