This question came up while going through the application of Eisenstein criterion: The $p$-th cyclotomic polynomial after changing the variable $x$ to $(x+1)$ satisfies Eisenstein criterion. That is the minimal polynomial of $\zeta_p-1$ is an Eisenstein polynomial.
Now let us take a general algebriac number field Q$[\alpha]=K$. Can one find another primitive element $\beta$ for $K$ such that its minimal polynomial is Eisenstein? As all quadratic fields arise from $\sqrt d$ which has equation $x^2-d$, it is true.
Since the evidence so far is from cyclotomic and quadratic, is this true for all abelian extensions?