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2 | Fixed the reference to Grothendieck Murre | ||
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Look at Lemma 2.1.3 vi.v) from Grothendieck and Murre: "The Tame Fundamental Group of a Formal Neighbourhood of a divsors with Normal Crossings on a Scheme". It says when given a tame field extension $L \supset K$, then its Galois closure will again be tame. Here, tameness is just defined with respect to one valution of $K$. But the proof should apply in your situation as well. |
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Look at Lemma 2.1.3 v) from Grothendieck and Murre: "The Tame Fundamental Group of a Formal Neighbourhood of a divsors with Normal Crossings on a Scheme". It says when given a tame field extension $L \supset K$, then its Galois closure will again be tame. Here, tameness is just defined with respect to one valution of $K$. But the proof should apply in your situation as well. |
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