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On the construction of normal wildly ramified extensions over Q p , (p ≠ 2)


Let p be an odd prime number, and let Q p 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 p . 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.

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Correspondence to Akram Lbekkouri.

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Lbekkouri, A. On the construction of normal wildly ramified extensions over Q p , (p ≠ 2). Arch. Math. 93, 331 (2009).

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  • Galois Group
  • Sylow Subgroup
  • Algebraic Closure
  • Minimal Polynomial
  • Newton Polygon