Let n > 0 be an odd integer. We show that every totally ramified extension of the 2-adic numbers of degree 2n has two automorphisms. When combined with Krasner’s Mass Formula, this allows us to count isomorphism classes of totally ramified 2-adic fields of degree 2n by discriminant. We further provide canonical defining Eisenstein polynomials for each extension. These results generalize previous work, which focused on the specific cases where n ≤ 7.

Eisenstein polynomials, p-adic fields, automorphism groups, totally ramified extensions.