Adele – c’est un nom si belle. (Oops, that’s bad French, isn’t it?)

I actually will not be able to finish the proof of the unit theorem here, because I don’t get to the ideles in this post. That will come next time (there are some of the same themes as here).

Let {K} be a global field, i.e. a finite extension of either {\mathbb{Q}} or {\mathbb{F}_p(t)}. Then we can consider the set absolute values on {K}. In the number field case, these are extensions (up to a power) of the archimedean absolute value on {\mathbb{Q}} or the {p}-adic absolute values by a theorem of Ostrowski classifying absolute values on {\mathbb{Q}}. In the function field case, we need another result.

Here’s how we define the adele ring. It is the restricted direct product

\displaystyle \mathbf{A}_K := \prod'_v K_v

where restricted means that any vector {(x_v)_{v \in V} \in \mathbf{A}_K} is required to satisfy {|x_v|_v \leq 1} for almost all {v}. This becomes a topological ring if we take a basis of the form

\displaystyle \prod_{v \in S} T_v \times \prod_{v \notin S} \mathcal{O}_v

where {T_v \subset k_v} are open and {\mathcal{O}_v} is the ring of integers, and {S} is a finite set containing the archimedean places. It is clear that addition and multiplication are continuous, and that {A_K} is locally compact. For {S} finite and containing the archimedean absolute values {S_\infty}, there is a subring {\mathbf{A}_K^S = \prod_{v \in S} K_v \times \prod_{v \notin S} \mathcal{O}_v}, and {\mathbf{A}_K} is the union of these subrings.

Since any {x \in K} is contained in {\mathcal{O}_v} for almost all {v} (this is analogous to a rational function on a curve having only finitely many poles), there is an injective homomorphism {K \rightarrow \mathbf{A}_K}.

Next, we may define a Haar measure on {\mathbf{A}_K^S} by taking the product of the Haar measures {\mu_v} on {K_v}, normalized such that {\mu_v(O_v )=1} for {v \notin S_\infty}. Thus one gets a (i.e., the) Haar measure on {\mathbf{A}_K} itself. (more…)

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The Artin-Whaples approximation theorem is a nice extension of the Chinese remainder theorem to absolute values, to which it reduces when the absolute values are discrete.

So fix pairwise nonequivalent absolute values {\left|\cdot\right|_1, \dots, \left|\cdot\right|_n} on the field {K}; this means that they induce different topologies, so are not powers of each other

Theorem 1 (Artin-Whaples)

Hypotheses as above, given {a_1, \dots, a_n \in K} and {\epsilon>0}, there exists {a \in K} with

 

\displaystyle \left|a - a_i\right|_i < \epsilon, \quad 1 \leq i \leq n.

  (more…)

Time to go back to basic algebraic number theory (which we’ll need for two of my future aims here: class field theory and modular representation theory), and to throw in a few more facts about absolute values and completions—as we’ll see, extensions in the complete case are always unique, so this simplifies dealing with things like ramification. Since ramification isn’t affected by completion, we can often reduce to the complete case. 

Absolute Values  

Henceforth, all absolute values are nontrivial—we don’t really care about the absolute value that takes the value one everywhere except at zero.

I mentioned a while back that absolute values on fields determine a topology. As it turns out, there is essentially a converse. 

Theorem 1 Let {\left|\cdot\right|_1}, {\left|\cdot\right|_2} be absolute values on {K} inducing the same topology. Then {\left|\cdot\right|_2} is a power of {\left|\cdot\right|_1}  (more…)

With the school year starting, I can’t keep up with the one-post-a-day frequency anymore. Still, I want to keep plowing ahead towards class field theory.

Today’s main goal is to show that under certain conditions, we can always extend valuations to bigger fields. I’m not aiming for maximum generality here though. 

Dedekind Domains and Extensions  

One of the reasons Dedekind domains are so important is

Theorem 1 Let {A} be a Dedekind domain with quotient field {K}, {L} a finite separable extension of {K}, and {B} the integral closure of {A} in {L}. Then {B} is Dedekind. (more…)

So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field {F} with an absolute value {\left \lvert . \right \rvert} like the rationals with the usual absolute value.

 Completions   

Definition 1 The completion {\hat{F}} of {F} is defined as the set of equivalence classes of Cauchy sequences:  (more…)

I was initially planning on doing a post on Hensel’s lemma. Actually, I think I’ll leave that for later, after I’ve covered some more number theory (which may motivate it better).

So the goal for the next several posts is to cover some algebraic number theory, eventually leading into class field theory. At least in the near future, I intend to keep everything purely local. Thus, the appropriate place to start is to discuss discrete valuation rings rather than Dedekind domains. 

Absolute Values  

Actually, it is perhaps more logical to introduce discrete valuations as a special case of absolute values, which in turn generalize the standard absolute value on {\mathbb{R}}(more…)