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The Artin-Whaples approximation theorem October 6, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
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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…)

Topologies determine the absolute value October 5, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
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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…)

Extensions of discrete valuations September 5, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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…)

Completions of fields September 1, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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…)

Discrete valuation rings and absolute values August 28, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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…)

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