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May 30 at 20:08 comment added R.P. I'm not an expert, but this question seems to split into two separate ones: first, finding out whether X becomes a self-product $Y^d$ over the algebraic closure of K for some suitable Y, which is a purely geometric question; second, finding out whether the Galois action on X is the correct one, which is an arithmetic question. The first question seems to be the more interesting one.
May 30 at 18:21 history edited Moinsdeuxcat CC BY-SA 4.0
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May 30 at 11:42 history edited Moinsdeuxcat CC BY-SA 4.0
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May 30 at 10:36 history asked Moinsdeuxcat CC BY-SA 4.0