I was going through my algebra homework and I realized we've never encountered a polynomial with roots with two orders of radicals, i.e. $(x-(3^{1/2}+3^{1/4})$ or something. Now, obviously, you could just multiply out from the factors, but that gives you some really nasty things. My question is: is this a coincidence? Are there certain conditions under which such roots form? Is there a way to prove that they can't form? Is it possible for "nice" polynomials, i.e. ones with rational coefficients?