Let p > 2 be prime. We study the problem of determining defining polynomials of totally ramified Galois extensions of the p-adic numbers of degree pⁿ for some positive integer n. By Local Class Field Theory, this amounts to finding one Eisenstein polynomial for each of the pⁿ isomorphism classes of cyclic extensions of degree pⁿ. The case n = 1 is due to Amano [1]. We give analogous results for the cases n = 2 and n = 3.

Eisenstein, p-adic, Galois, totally ramified, cyclic.