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. When p n, all extensions are tamely ramified and are well understood [Jones and Roberts 2006]. Likewise, when p = n, the situation has been solved since the early 1970s [Amano 1971; Jones and Roberts 2006]. The difficult cases where p | n and n is composite have been dealt with on a case-by-case basis for low degrees n and small primes p.

Let K be a Galois extension of the 2-adic numbers Q 2 of degree 16 and let G be the Galois group of K/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/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 Q 2 is G.

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