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

C Awtrey,J Johnson,B Sinclair,J Milstead

doi:10.12732/ijpam.v103i4.13

C Awtrey, J Johnson + Show 2 more

Open Access

https://doi.org/10.12732/ijpam.v103i4.13

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Abstract

Let K be a Galois extension of the 2-adic numbers Q2 of degree 16 and let G be the Galois group of K/Q2. 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/Q2 except for E16, 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 Q2 is G.

Highlights

Two important problems in Galois theory are the following.Received: August 8, 2015 §Correspondence author c 2015 Academic Publications, Ltd. url: www.acadpubl.euC
This paper focuses on solving problems (P1) and (P2) for the case of Galois extensions of degree 16 defined over the 2-adic numbers
As a corollary of our work we prove that the group E16 does not occur as the Galois group of a degree 16 2-adic field

Summary

Introduction

Two important problems in Galois theory are the following. Received: August 8, 2015 §Correspondence author c 2015 Academic Publications, Ltd. url: www.acadpubl.eu. Since the number of degree n extensions of Qp is finite [14], it is possible to determine defining polynomials for all extensions along with their corresponding Galois groups If we tabulate these results, we effectively provide an answer for (P2) in the cases where a solution exists. The situation is similar when p = n, with [1] providing all pertinent theory In both cases, explicit methods for the computation of defining polynomials and Galois groups can be found in [12]. This paper focuses on solving problems (P1) and (P2) for the case of Galois extensions of degree 16 defined over the 2-adic numbers. We give an algorithm for constructing quadratic extensions of 2-adic fields in general and use our method to produce polynomials whose Galois groups correspond to the remaining 13 transitive subgroups of S16 of order 16

Computing Galois Groups

Defining Polynomials for Octic 2-adic Fields

Quadratic Extensions of Octic 2-adic Fields

Polynomials