Abstract:
Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results.
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Received: 27 October 1998 / Revised version: 3 February 1999
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Jensen, C., Prestel, A. How often can a finite group be realized as a Galois group over a field?. manuscripta math. 99, 223–247 (1999). https://doi.org/10.1007/s002290050171
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DOI: https://doi.org/10.1007/s002290050171