6
$\begingroup$

Background and Motivation

Local Class Field Theory says that abelian extensions of a finite extension $K/\mathbb{Q}_p$ are parametrized by the open subgroups of finite index in $K^\times$. The correspondence takes an abelian extension $L/K$ and sends it to $N_{L/K}(L^\times)$, and this correspondence is bijective.

If one starts instead with a galois extension $L/K$ that isn't abelian, one can then ask "What abelian extension does $N_{L/K}(L^\times)$ correspond to?" The answer is the maximal abelian extension of $K$ contained in $L$.


The hypothesis of being galois isn't necessary in the statement of the non-abelian theorem: both the question and the answer still make sense. I am thus asking

Assume that $L/K$ as above. Is the abelain extension of $K$ corresponding to $N_{L/K}(L^\times)$ the maximal abelian extension of $K$ contained in $L$?

A couple examples to illustrate this problem (including the example that I was told would sink this):

If $p > 2$, then consider $L = \mathbb{Q}_p(\sqrt[p]{p})$. The norm subgroup that I am anticipating is all of $\mathbb{Q}_p^\times$. Moving into the galois closure and using the theorem, one gets that $N_{L/\mathbb{Q}_p}(L^\times)$ contains $p^\mathbb{Z} \times (1 + p\mathbb{Z}_p)$. Moreover, the norm of a number $a \in \mathbb{Z}_p$ will just be $a^p$, which is congruent to $a$ mod $p$, so the norm will also hit something congruent to any given root of unity in $\mathbb{Q}_p$, and that was all that I was missing from the earlier note.

One can also, using the same idea (moving into the galois closure and getting a lot of information from that, and then using the explicit structure of the field that one started with) show that this works for $K(\sqrt[n]p)$ with $(n, p^{f(K/\mathbb{Q}_p)}) = (n,p) = 1$ as well.

$\endgroup$
4
$\begingroup$

Yup, it is. This is Theorem III.3.5 (norm limitation theorem) of Milne's Class field theory notes (available here). The global analogue is Theorem VIII.4.8.

$\endgroup$
6
$\begingroup$

As an addition to the answers given above: these theorems were first formulated (and proved) by Arnold Scholz in

  • Die Abgrenzungssätze für Kreiskörper und Klassenkörper (Limitation theorems for cyclotomic fields and class fields), Sitzungsberichte Akad. Berlin 1931, 417 - 426
$\endgroup$
2
$\begingroup$

Hi Erick. Yes, this is true. Maybe it's in Iwasawa's Local Class Field Theory? For the global analogue, see the last exercise in Cassels & Frohlich.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.