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?

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    $\begingroup$ Every root of an Eisenstein polynomial of degree $n$ for a prime $p$, has extended $p$-adic valuation $\frac{1}{n}$. So the resulting number field is totally ramified, and this is a necessary condition. I don't know the inverse. $\endgroup$ – Mostafa Feb 20 '15 at 9:17

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 $n$).

  • $\begingroup$ Thanks for a clear answer, Michael Stoll, and also to Mostafa. $\endgroup$ – P Vanchinathan Feb 20 '15 at 15:31

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