IJPAM: Volume 75, No. 2 (2012)

DIHEDRAL $p$-ADIC FIELDS OF PRIME DEGREE

Chad Awtrey$^1$, Trevor Edwards$^2$
$^{1,2}$Department of Mathematics and Statistics
Elon University
Campus Box 2320, Elon, NC 27244, USA


Abstract. 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 $D_n$, the dihedral group of order $2n$. If $p\nmid n$, the extensions are tamely ramified and are straightforward to classify; there is a unique such extension if $n\mid p+1$ 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.

Received: July 8, 2011

AMS Subject Classification: 11S05, 11S15, 11S20, 20B35

Key Words and Phrases: $p$-adic, extension fields, Galois group, dihedral, inertia, ramification

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 75
Issue: 2