what's the rationality of the roots of polynomial related to the rationality of its coefficient? Here is the question just came up to me, is that true for a polynomial with irrational coefficient cannot has a rational root. Is any theorem about this?
As another example, the polynomial $$ x^2 - (\pi +1) x + \pi $$ has $1$ as a root, and no multiple of it by a nonzero constant has all coefficients rational.
If you want all coefficients to be non-rational, (see comment by @MartinBrandenburg) $$ \pi x^2 - (\pi^2 +\pi) x + \pi^2 $$ will do.