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

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

Dedekind domains August 31, 2009

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