IJPAM: Volume 103, No. 4 (2015)

GROUPS OF ORDER 16 AS GALOIS GROUPS
OVER THE 2-ADIC NUMBERS

Chad Awtrey$^1$, John Johnson$^2$, Jonathan Milstead$^3$, Brian Sinclair$^4$
$^{1,2}$Department of Mathematics and Statistics
Elon University
Campus Box 2320
Elon, NC 27244, USA
$^{3}$Department of Mathematics and Statistics
University of North Carolina
116 Petty Building, 317 College Ave
Greensboro, NC 27412, USA


Abstract. Let $K$ be a Galois extension of the 2-adic numbers $\mathbf{Q}_2$ of degree 16 and let $G$ be the Galois group of $K/\mathbf{Q}_2$. We show that $G$ can be determined by the Galois groups of the octic subfields of $K$. We also show that all 14 groups of order 16 occur as the Galois group of some Galois extension $K/\mathbf{Q}_2$ except for $E_{16}$, the elementary abelian group of order $2^4$. For the other 13 groups $G$, we give a degree 16 polynomial $f(x)$ such that the Galois group of $f$ over $\mathbf{Q}_2$ is $G$.

Received: August 8, 2015

AMS Subject Classification: 20B35, 12F10, 11S20

Key Words and Phrases: groups of order 16, extension fields, Galois group, 2-adic

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DOI: 10.12732/ijpam.v103i4.13 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 103
Issue: 4
Pages: 781 -


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