#### Vol. 8, No. 2, 2015

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### Chad Awtrey, Nicole Miles, Jonathan Milstead, Christopher Shill and Erin Strosnider

Vol. 8 (2015), No. 2, 329–336
##### Abstract

We study the 590 nonisomorphic degree 14 extensions of the 2-adic numbers by computing defining polynomials for each extension as well as basic invariant data for each polynomial, including the ramification index, residue degree, discriminant exponent, and Galois group. Our study of the Galois groups of these extensions shows that only 10 of the 63 transitive subgroups of ${S}_{14}$ occur as a Galois group. We end by describing our implementation for computing Galois groups in this setting, which is of interest since it uses subfield information, the discriminant, and only one other resolvent polynomial.

##### Keywords
2-adic, extension fields, Galois group, local field
##### Mathematical Subject Classification 2010
Primary: 11S15, 11S20