Irrational roots of polynomials with integer coefficients

Question

Consider an equation of the form $$ p(x) = 0, $$ where $p(x) = \sum_{i=0}^{n} a_i x^i$ and all $a_i \in \mathbb{Z}$. Is there an algorithm which can tell us that this equation has a root of the form $a + b \sqrt{n}$ for a given $n$ (i.e. $a + b\sqrt{2}$) with $a,b \in \mathbb{Q}$? The only thing I can think of is to divide the polynomial by $(x-a)^2 - nb^2$, and then obtain some condition on the remainder (which must be linear and equal to zero) to find $a$ and $b$.