# Adjoining an element with given minimal polynomial to a DVR of characteristic p.

Let $$A$$ be a DVR of characteristic $$p$$, with $$\pi$$ a uniformising parameter, with $$K=frac(A)$$ the field of fractions. Consider the extension $$L=K(\alpha)$$ where $$\alpha$$ has minimal polynomial $$y^p+\pi^b y+\pi^c$$, with $$0. The problem is to show the existence of an element of $$L$$ with minimal polynomial $$y^p+u\pi y+v\pi^{p(c-b)}$$, with $$u,v\in A^*$$.

I think this is going to be manipulating the known minimal polynomial of $$\alpha$$ to produce such another element, eg $$\alpha \pi^k+\pi^l$$, for $$k,l\in\mathbb{Z}$$, possibly using the Frobenius map.

I wasn't able to make this work however, so it may require a more sophisticated idea. Any hints would be much appreciated.

• I think your attempt is good. For the simplest case $b=c=1$, setting $\alpha+1$ as the new element works. For the general case, I would try $k=l :=c-b$, in other words, the new element would be $\pi^{c-b} \cdot(\alpha+1)$. – Torsten Schoeneberg Dec 4 '18 at 2:01