TITLE: Do all algebraic number fields arise from Eisenstein polynomials?
QUESTION [6 upvotes]: 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?

REPLY [13 votes]: 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 $n$).