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Totally ramified extensions October 23, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
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Today we consider the case of a totally ramified extension of local fields {K \subset L}, with residue fields {\overline{K}, \overline{L}}—recall that this means {e=[L:K]=n,f=1}. It turns out that there is a similar characterization as for unramified extensions. (more…)

Unramified extensions October 20, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
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As is likely the case with many math bloggers, I’ve been looking quite a bit at MO and haven’t updated on some of the previous series in a while.

Back to ANT. Today, we tackle the case {e=1}. We work in the local case where all our DVRs are complete, and all our residue fields are perfect (e.g. finite) (EDIT: I don’t think this works out in the non-local case). I’ll just state these assumptions at the outset. Then, unramified extensions can be described fairly explicitly. (more…)

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…)

A talk on the p-adic numbers September 16, 2009

Posted by Akhil Mathew in algebraic number theory, General, math education, number theory.
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The start of the academic year has made it much more difficult for me to get in serious posts as of late, and the number theory series has slowed.  Things should clear up at least somewhat in a few more weeks.   In the meantime, I’ll do something that occurred to me a while back but I then forgot about: posting a talk.

I took an independent study course last semester on class field theory.  As is traditional, I gave a talk last May after the course on some aspects of the subject matter.  Several faculty members at the university and teachers in my school attended, along with some undergraduates there.  In the talk, I gave an elementary overview of the p-adic numbers, assuming no more than basic number theory and point-set topology.

Anyway, I am posting the (slightly corrected) presentation and the notes here.

e, f, and the remainder theorem September 12, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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So, now to the next topic in introductory algebraic number theory: ramification. This is a measure of how primes “split.”  (No, definitely wrong word there…)

e and f 

Fix a Dedekind domain {A} with quotient field {K}; let {L} be a finite separable extension of {K}, and {B} the integral closure of {A} in {L}. We know that {B} is a Dedekind domain.

(By the way, I’m now assuming that readers have been following the past few posts or so on these topics.)

Given a prime {\mathfrak{p} \subset A}, there is a prime {\mathfrak{P} \subset B} lying above {\mathfrak{p}}. I hinted at the proof in the previous post, but to save time and avoid too much redundancy I’ll refer interested readers to this post.

Now, we can do a prime factorization of {\mathfrak{p}B \subset B,} say {\mathfrak{p}B = \mathfrak{P}_1^{e_1} \dots \mathfrak{P}_g^{e_g}}. The primes {\mathfrak{P}_i} contain {\mathfrak{p}B} and consequently lie above {\mathfrak{p}}. Conversely, any prime of {B} containing {\mathfrak{p}B} must lie above {\mathfrak{p}}, since if {I} is an ideal in a Dedekind domain contained in a prime ideal {P}, then {P} occurs in the prime factorization of {I} (to see this, localize and work in a DVR). (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…)

Hensel’s lemma and a classification theorem September 2, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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So, I’ll discuss the proof of a classification theorem that DVRs are often power series rings, using Hensel’s lemma. 

Systems of representatives  

Let {R} be a complete DVR with maximal ideal {\mathfrak{m}} and quotient field {F}. We let {k:=R/\mathfrak{m}}; this is the residue field and is, e.g., the integers mod {p} for the {p}-adic integers (I will discuss this more later).

The main result that we have today is:

Theorem 1 Suppose {k} is of characteristic zero. Then {R \simeq k[[X]]}, the power series ring in one variable, with respect to the usual discrete valuation on {k[[X]]}. (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…)

Dedekind domains August 31, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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2 comments

Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of integers in any finite extension of {\mathbb{Q}} (i.e. number fields). In these, you don’t necessarily have unique factorization. But you do have something close, which makes these crucial. 

Definition 1 A Dedekind domain is a Noetherian integral domain {A} that is integrally closed, and of Krull dimension one—that is, each nonzero prime ideal is maximal.   (more…)

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