Let p be an odd prime number, and let Q
be the field of rational p-adic numbers.The aim of this work is the determination of the standard form of an Eisenstein polynomial defining a normal wildly ramified extension of Q
. We prove first the equivalence between normality and cyclicity, give some essential normality conditions for the general case (degree p
n), then we solve the problem completely for the case (degree p
2) also, we obtain that the normality depends on seven congruences modulo p
m between the coefficients of the considered polynomial with just m = 2 or 3. Note that the case (degree p) was solved by Öystein Ore (see Math. Annalen 102 (1930), 283–304). Also examples are given.