mathworld for one that's typeset properly
consider (x - sqrt(2)) * (x + sqrt(2)). Not sure how to handle 2^(1/3) and 3+2^(1/2) but someone else will probably jump on those.
For the second part see here
Edit: I see from mathworld that the radical integers are a subring of the algebraic integers. If you wanted to prove that much stronger result then all the special cases asked for in the problem would immediately follow.