DIHEDRAL $p$-ADIC FIELDS OF PRIME DEGREE

Let p and n be odd prime numbers. We study degree n extensions of the p-adic numbers whose normal closures have Galois group equal to Dn, the dihedral group of order 2n. If p ∤ n, the extensions are tamely ramified and are straightforward to classify; there is a unique such extension if n | p+ 1 and none otherwise. If p = n, we follow Amano and show there are six such extensions if p = 3 and three otherwise. For each extension, we provide a defining polynomial and compute its inertia subgroup.