Adjoining an element with given minimal polynomial to a DVR of characteristic p.
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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 $0<bleq c$ . The problem is to show the existence of an element of $L$ with minimal polynomial $y^p+upi y+vpi^{p(c-b)}$ , with $u,vin A^*$ . 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 $k,linmathbb{Z}$ , 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.
abstract-algebra algebraic-number-theory valuation-theory local-rings
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