x^7 is a polynomial whose roots can be found using only algebraic operations. Send the Fields Medal to the University of Ottawa, care of pboily.
Seriously, the big deal with polynomials of degree > 4 is that there is no general "algebraic" way that will allow you to find the roots of all such polynomials (thank you, Abel and Galois). But you may still find the roots of particular polynomials using only algebraic operations.