TITLE: Does one real radical root imply they all are?
QUESTION [50 upvotes]: Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?

REPLY [2 votes]: WARNING: Wrong answer follows, as per comments. Kept here to deter others from making the same mistake. 
Perhaps $$x_1=\sqrt{2+2^{1/4}}+\sqrt{2-2^{1/4}},\quad x_2=\sqrt{2+i2^{1/4}}+\sqrt{2-i2^{1/4}}$$ are real numbers, solutions of $\bigl((x^2-4)^2-16\bigr)^2=32$, one expressed using real radicals, the other, not.