Suppose $ (L/L') $ is an galois extension , where both fields are extension of $Q$ of $\dim n $ and $\dim n^{'}$ respectively.Suppose we consider the norm map $ Nr_{(L/L')} :L \rightarrow L' $. Suppose $ U $ be a subspace of L' of $\dim n'-1 $ then what will be the maximum dimensional subspace in $L$ such that norm of every elements belong to this subspace ? Can we say anything about the pre-image of this subspace?
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$\begingroup$ I would be surprised if there were a reasonable answer to this question. You are mixing a multiplicative map, highly non linear (the norm), which something additive (subspaces of $L$ and $L'$) . Could you please explain the context and the motivation? Did you work out some examples ? $\endgroup$– GreginGreJan 6, 2022 at 9:08
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$\begingroup$ Actually if we take finite field then the norm map is subjective but I don't have any idea about the subjectivity condition valid in $\mathbb Q $ or some extension. $\endgroup$– SkyJan 6, 2022 at 9:45
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$\begingroup$ If i remember properly, ihe norm map is never surjective for number fields, but i don't see the relation with your question, since you are asking about the values abutting into a hyperplane. What is really your question? Once again, what is the context/motivation? did you work out examples ? $\endgroup$– GreginGreJan 6, 2022 at 9:47
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$\begingroup$ I have mentioned hyperplane because I thought norm map is surjective , I thought as trace map, but I know that norm map is not a linear. Here you are saying already that subjectivity never happens in number fields. So I think it has no specific answer.. Anyway can you provide me the topic where I can found norm map is not surjective in number field $\endgroup$– SkyJan 6, 2022 at 9:54
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$\begingroup$ Then actually the example of non surjective map,helps us to finding the counter examples. $\endgroup$– SkyJan 6, 2022 at 10:00
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