I have a question about Teiji Takagi's original motivation for his groundbreaking reformulation of class field theory implementing the ideal norms $N_{\mathfrak{m}}(L/K)$ as new "key players" in his new formulation, This approach paves more or less the way for Artin's modern formulation of CFT adapting this approach using norms, even giving structure of certain profinite-topology determined by these.
Let's recapitulate shortly what happened roughly before Takagi came up with his groundbreaking reformulation of CFT in terms of norms (for details see Keith Conrad's Notes on history of class field theory:
In his 1891 book on elliptic functions and algebraic numbers, Heinrich Weber introduced the label “class field” studying Abelian field extensions $L/K$ in terms of certain group, the ideal class group $I$, so at first a class field was just a particular abelian (conjecturally unramified) extension of an imaginary quadratic field, whose Galois group is isomorphic to the ideal class group of the base field. In 1897, Weber extended the concept of ideal class group: for any number field $K$ and nonzero ideal $\mathfrak{m}$ in $\mathcal{O}_K$, let $I_{\mathfrak{m}}$ be the group of fractional ideals in $K$ relatively prime to modulus $\mathfrak{m}$ and let $P^+_{\mathfrak{m}}$ be the group of principal fractional ideals $(\alpha/\beta)$ where $\alpha$ and $\beta$ are nonzero integers in $K$ subjected to certain additional conditions (see above linked source for details)
Any intermediate group
$$ P^+_{\mathfrak{m}} \subset H \subset I_{\mathfrak{m}}$$
is called an ideal group with modulus $\mathfrak{m}$ and the quotient $I_{\mathfrak{m}}/H $ is called a generalized ideal class group.
Takagi introduced the intermediate group $H_{\mathfrak{m}}(L/K):= P_{\mathfrak{m}} N_{\mathfrak{m}}(L/K)$
where $P_{\mathfrak{m}} $ is slight modification of $P^+_{\mathfrak{m}}$ and
$$ N_{\mathfrak{m}}(L/K):= \{\mathfrak{a} \in I_{\mathfrak{m}} \ | \ \mathfrak{a} =N(\mathfrak{A}) \text{ for some fractional ideal in } L \text{ coprime to } \mathfrak{m} \} $$
Question: Obviously Takagi's key motivation was to find a kind of "canonical" group lying between $ P^+_{\mathfrak{m}} $ and $I_{\mathfrak{m}}$, therefore the factor $P_{\mathfrak{m}} $ ( up to the $+$-sign issue which is about adding infinite places and I would like to sweep this under the rug in this discussion) in $H_{\mathfrak{m}}(L/K)$ seems to be "reasonable". But I not see why Takagi decided to implement the norm factor $ N_{\mathfrak{m}}(L/K)$ in his definition.
It seems to "be fallen from heaven" without any conceptional idea in background. Just, like a clever guess, and my question is simply if there exist a concrete reason for this choice. (Note that in this time there was no theory of adeles and their profinite topology defined by norm groups known).
ADDENDUM: Here I found historical notes on Takagi's works where is briefly remarked that Takagi was highly influenced by Hilbert's works on unramified class field theory where he gave a new reciprocity law for the so called Hilbert's norm residue symbol (see here for the definition). I'm not sure if - beside the same name- Takagi's approach using norms is somehow inspired by these norm residue symbols introduced by Hilbert. Maybe there is a concrete example calculation in Hilbert's works involving norm ideals, which might had inspired Takagi to abstract it and build his theory on it, I don't know.
Honestly, I not see any mathematical connection, between these two objects - the ideal norm and Hilbert's norm residue - beside similar names as I remarked before. Does anybody see the connection (... even about any vague one I would be happy; it's really about what might had inspired Takagi to use this norm terminology)
Also I'm not completely sure if it make sense to post this question here or in MathStackExchange.
