Konstruktion algebraischer Zahlk�rper mit beliebiger Gruppe von Primzahllpotenzordnung I

Algebraic Number Theory

“…If Jl.1 cj: K this is not difficult to show (Scholz (1937), Reichardt (1937». In the case Jl.1 c K one proceeds as follows. If Jl.1 cj: K this is not difficult to show (Scholz (1937), Reichardt (1937». In the case Jl.1 c K one proceeds as follows.…”

“…If then f: E ~ G is a morphism of an arbitrary profinite group E, we obtain the diagrams The embedding problem with given local behavior asks for morphisms tP which induce given local morphisms tPv for certain places v of K. The proof uses ideas of Scholz (1937) and Reichardt (1937) and Galois cohomology of algebraic number fields. Let G be a finite solvable group and K an algebraic number field.…”

Nilpotent extensions of number fields with bounded ramification

“…Scholz [18] based in Brauer's work gave a sufficient condition for the resolution of this central embedding problem in terms of the extension. This condition is given in the following definition.…”

“…This is the Inverse Problem of Galois Theory. When such extension exists we say that G is realizable over Q. Scholz [18] and independently Reichardt [12] proved that if G is a finite p-group, p an odd prime, then G is realizable over Q. They used a criterion given by Brauer [2].…”

Notes on the minimal number of ramified primes in some l-extensions of Q

“…Scholz [11] and Reichardt [9] proved that every lgroup G can be realized as the Galois group of some extension M of the rational number field Q. We are interested in the number of primes which are ramified in M/Q.…”