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.


1 Answer 1


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

These two uses of the word norm are definitely related.

On a principal ideal $(\alpha)$ in the integers of a number field $K$, its ideal norm ${\rm N}((\alpha)) := |\mathcal O_K/(\alpha)|$ is its field norm in absolute value: ${\rm N}((\alpha)) = |{\rm N}_{K/\mathbf Q}(\alpha)|$. This is the entire reason we give the size of a residue ring ${\mathcal O}_K/\mathfrak a$ the name "ideal norm".

Hilbert's norm residue symbol is trivial on values of the field norm, just as the Legendre symbol is also called a quadratic residue symbol because it is trivial on quadratic residues.

  • $\begingroup$ so on level of principal ideals the generators of such norm ideals form the radical - or at least loosely speaking a kind of ( in sense of theory of bilinear forms) - of Hilberts norm residue symbol? And maybe Takagi intended to generlize the notion of norm ideals with respect ideals generated by more than one generator (ie non principal ones), but there exist no "extended gadget" analogous to Hilbert residue symbol for non principal ideal with propery that norm ideals are exactly those beeing trivial on it. So Takagi maybe defined ideal norms "directly" without recognizing them as kind of $\endgroup$
    – user267839
    Aug 13, 2023 at 7:27
  • $\begingroup$ a "radical" with respect a hypothetical extension of Hilberts residue norm symbol to non principal ideals? Does this "interpretation" make sense? $\endgroup$
    – user267839
    Aug 13, 2023 at 7:29
  • $\begingroup$ *add: name "radical" in comment above was bad choosen. Better say just that gen's of principal ideal lie in "kernel" of norm symbol $\endgroup$
    – user267839
    Aug 13, 2023 at 8:06
  • 1
    $\begingroup$ Hilbert symbols are related to the reciprocity isomorphism in local class field theory, so an "extended gadget" on number fields analogous to Hilbert symbols is arguably the Artin map in global class field theory, and the Artin map did not exist when Takagi did his work: Takagi's proof of an isomorphism between global abelian Galois groups and generalized ideal class groups did not involve a direct definition of an isomorphism with nice functorial properties. $\endgroup$
    – KCd
    Aug 13, 2023 at 16:57
  • $\begingroup$ Do you know the title of this paper where Takagi propes this iso between isomorphism between global abelian Galois groups and generalized ideal class groups? What I'm wondering about is that in the second source linked above is several times emphasised how stronly Hilbert's works on unramified case influenced Takagi. And the question is which techniques Takagi adapted in crucial way for his proof of this statement from Hilberts works? $\endgroup$
    – user267839
    Aug 14, 2023 at 21:56

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