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 algebraic 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? 



Do all algebraic number fields arise from Eisenstein polynomials?
As already indicated by the comment of Mostafa, the criterion is that $K$ is totally ramified at some prime $p$. Mostafa's comment shows that this is necessary. To see that it is also sufficient,
take any integral element $\alpha$ of $K$
whose $p$-adic valuation is $1/n$ (if the valuation is normalizied to be 1 at $p$),
where $n = [K : \mathbb Q]$; then the minimal polynomial of $\alpha$ will be
a $p$-Eisenstein polynomial (of degree $n$).


