Abstract
Let K be a field, and let f:X→Y be a finite étale cover between reduced and geometrically irreducible K-schemes of finite type such that fKs is Galois. Assuming f admits a Galois K-form f¯:X¯→Y, we use it to analyze fields of definition over K for the Galois property of f and the presence of K-points in general K-forms f′:X′→Y over Y(K).Additionally, we show that if K is Hilbertian, the group G is non-abelian, and the base variety is rational, then there are finite separable extensions L/K such that some L-form of fL does not descend to a cover of Y.
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