Let $K/P$ be a finite extension of number fields and $\epsilon_{K/P}:[\mathfrak{a}] \in Cl(P) \rightarrow [\mathfrak{a}.\mathcal{O}_K]\in Cl(K)$ be the ideal class transfer homomorphism. It's well known that there exists an isomorphism $\gamma:Cl(K) \rightarrow Gal(H_K/K)$ by $[\mathfrak{b}] \mapsto \left( \frac{H_K/K}{\mathfrak{b}}\right)$ (the Artin symbol at $\mathfrak{b}$), for the Hilbert class field $H_K$ of K. Is there any result (s) about the combination $\gamma \circ \epsilon_{K/P}$?
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1$\begingroup$ This is known as the "Verlagerung" map (German for "transfer"). It doesn't have a particularly nice description purely on the Galois side. If you google for "class field theory" and "Verlagerung" you'll find plenty of references. $\endgroup$– David LoefflerCommented May 27, 2020 at 8:55
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$\begingroup$ Many thanks for your response.Using the "transfer map", I want to see what happens in $P \subseteq F \subseteq K$ a tower of finite extensions of number fields. Roughly speaking, what is the image of "combination" of transfer maps? $\endgroup$– A. MaarefparvarCommented May 27, 2020 at 13:18
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1$\begingroup$ Harvey Cohn has written repeatedly about transfer maps in class field theory in his books on algebraic number theory and the construction of class fields. $\endgroup$– Franz LemmermeyerCommented Jun 25, 2020 at 14:03
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