Abstract

We study a variant of the inverse problem of Galois theory and Abhyankar’s conjecture. If p is an odd rational prime and G is a finite p-group generated by s elements, s minimal, does there exist a normal extension L ∕ Q such that Gal (L ∕ Q)≅G with at most s rational primes that ramify in L? Given a nilpotent group of odd order G with s generators, we obtain a Galois extension L ∕ Q with precisely s prime divisors of Q ramified. Furthermore if K is a number field satisfying K ∩ Q(ζ_piⁿⁱ) = Q for each rational prime p_i, such that p_i^n_i|∘ (G), p_i^n_i+1| ∕ ∘ (G), and such that there exists a rational prime q inert in K ∕ Q, we obtain a Galois extension E ∕ K with precisely s prime divisors of K ramified. An adaptation of the Scholz-Reichardt method for the embedding problem is our main tool.

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