It is my understanding that the Abel-Ruffini Theorem implies that certain polynomial equations $(x^5-x+1=0$, for instance) have transcendental roots. However, the Fundamental Theorem of Algebra states that we can factorise any polynomial into quadratics, and we can then solve these with the quadratic equation (implying that the solution would be algebraic).
On second thought, the only way I can see this to be resolved is if the quadratic factors of our polynomials have transcendental coefficients. Is this the case?