#### Vol. 14, No. 3, 2021

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Eisenstein polynomials defining Galois dihedral $p$-adic fields of degree $2p$

Vol. 14 (2021), No. 3, 463–474
##### Abstract

Let $p>2$ be prime, ${ℚ}_{p}$ the field of $p$-adic numbers, and $f\left(x\right)\in {ℚ}_{p}\left[x\right]$ an Eisenstein polynomial of degree $2p$. We give necessary and sufficient conditions on the coefficients of $f\left(x\right)$ for its Galois group to be isomorphic to the dihedral group of order $2p$.

##### Keywords
Eisenstein polynomials, $p$-adic fields, dihedral, totally ramified extensions
##### Mathematical Subject Classification
Primary: 11S05, 11S15, 11S20