On the index of generalized trinomials over Valued fields

Abstract

Let $\nu$ be a Krull valuation of arbitrary rank on a field with valuation ring $R_\nu$, and let $\theta$ be a root of the irreducible polynomial $F(x)=(x^k+c)^m-ax^n\in R_\nu[x]$ and $1\leq n<km$. We establish necessary and sufficient conditions for the integral closedness of $R_\nu[\theta]$, expressed explicitly in terms of the coefficients $a$, $b$, and the integers $m$, $n$, and $k$. In particular, when $\nu$ is the $p$-adic valuation on $\mathbb{Q}$, our results yield criteria for determining the primes dividing the index $[\mathbb{Z}_K:\mathbb{Z}[\theta]],$ where $K=\mathbb{Q}(\theta)$ and $\mathbb{Z}_K$ denotes the ring of integers of $K$.