  We present algorithmic, complexity and implementation results for the problem
of isolating the real roots of a univariate polynomial in $B_{\alpha} \in
L[y]$, where $L=\QQ(\alpha)$ is a simple algebraic extension of the rational
numbers. We consider two approaches for tackling the problem. In the first
approach using resultant computations we perform a reduction to a polynomial
with integer coefficients. We compute separation bounds for the roots, and
using them we deduce that we can isolate the real roots of $B_{\alpha}$ in
$\sOB(N^{10})$, where $N$ is an upper bound on all the quantities (degree and
bitsize) of the input polynomials. In the second approach we isolate the real
roots working directly on the polynomial of the input. We compute improved
separation bounds for real roots and we prove that they are optimal, under mild
assumptions. For isolating the roots we consider a modified Sturm's algorithm,
and a modified version of \func{descartes}' algorithm introduced by Sagraloff.
For the former we prove a complexity bound of $\sOB(N^8)$ and for the latter a
bound of $\sOB(N^{7})$. We implemented the algorithms in \func{C} as part of
the core library of \mathematica and we illustrate their efficiency over
various data sets. Finally, we present complexity results for the general case
of the first approach, where the coefficients belong to multiple extensions.
