Nilpotent Extensions of Number Fields with Bounded Ramification
نویسندگان
چکیده
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(ζpni i ) = Q for each rational prime pi, such that p ni i | ◦ (G), pi i | /◦(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 ScholzReichardt method for the embedding problem is our main tool.
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