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I'm taking a class in CFT and I'm trying to figure out what the theorems say and what they can be used for to get a "feel" for them. More explicitly, say I take $\mathbb{Q}_p$, so we have the local Artin homomorphism:

$\theta:\mathbb{Q}_p^\times \to \textrm{Gal}(\mathbb{Q}_p^{ab}/\mathbb{Q}_p)$.

This map is only an isomorphism if the take the profinite completion of $\mathbb{Q}_p^\times$ on the left, but otherwise we have a bijection between the finite-index open subgroups of $\mathbb{Q}_p^\times$ and abelian extensions of $\mathbb{Q}_p$.

Let's say we take the subgroup $\mathbb{Q}_p^{\times 2}$ of $\mathbb{Q}_p^\times$ and for simplicity let's assume that $p\neq 2$. Can we actually somehow explicitly write down the abelian extension that this subgroup corresponds to? None of the theorems look constructive, but because they are considered so useful, I guess there has to be ways of writing down the corresponding abelian extension explicitly through generators?

I still need to look into profinite groups, but my understanding is that for the previous example the order of the abelian extension corresponding to $\mathbb{Q}_p^{\times 2}$ has order $(\mathbb{Q}_p^\times:\mathbb{Q}_p^{\times 2})$.

If this is possible, does anyone know of a source where I could find examples of this?

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Explicit class field theory over local fields is well known and described by the Lubin-Tate theory. See Iwasawa's 'Local Class Field Theory' or Chapter 1 of Milne's 'Class Field Theory' (jmilne.org/math/CourseNotes/cft.html) for reference. –  jspecter Sep 22 '11 at 14:38
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1 Answer 1

As jspecter's comment says, the general case of explicit class field theory over local fields is given by Lubin-Tate theory. But in your particular case of $\mathbb{Q}_p^{\times 2}$ in $\mathbb{Q}_p^{\times}$, we can work it out by hand. For simplicity, I will assume that $p$ is odd.

The squares in $\mathbb{Q}_p^{\times}$ are elements of the form $p^{2n} u$ where $u$ is an element of $\mathbb{Z}_p$ which congruent to a non-zero square modulo $p$. As a result, $(\mathbb{Q}_p^{\times} : \mathbb{Q}_p^{\times 2}) \cong C_2 \times C_2$ (the direct product of two cyclic groups of order 2). By class field theory, the corresponding abelian extension of $\mathbb{Q}_p$ must have Galois group $C_2 \times C_2$. What could it be?

Well, there's really only one way to get elements of order 2 in a Galois group: adjoin square roots, and there aren't too many square roots we can adjoin that aren't already in $\mathbb{Q}_p$. In fact, a set of coset representatives for $\mathbb{Q}_p^{\times 2}$ in $\mathbb{Q}_p^{\times}$ is given by $\{1,s,p,sp\}$ where $s$ is any element of $\mathbb{Z}_p$ which is a non-square modulo $p$. So we should try the field $K = \mathbb{Q}_p(\sqrt{s}, \sqrt{p})$, which does have the right Galois group $C_2 \times C_2$.

Is $K$ the right field? Yes, it is, because, as you can easily check, $\mathbb{Q}_p^{\times 2}$ is the unique subgroup of $\mathbb{Q}_p^{\times}$ whose quotient group is isomorphic to $C_2 \times C_2$. Therefore, by class field theory, there is a unique abelian extension with Galois group $C_2 \times C_2$, so $K$ is the field we are looking for.

Some other cases you can think about:

  1. There are 3 subgroups of index 2 between $\mathbb{Q}_p^{\times 2}$ and $\mathbb{Q}_p^{\times}$ ($p$ odd). What are the extensions corresponding to each of them?

  2. What happens when $p=2$ ? What is the extension corresponding to $\mathbb{Q}_2^{\times 2}$? What about all the subgroups in between $\mathbb{Q}_2^{\times 2}$ and $\mathbb{Q}_2^{\times}$?

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