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Totally ramified extensions
*October 23, 2009*

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

Tags: discrete valuation rings, Eisenstein polynomials, ramification, totally ramified extensions

6 comments

Tags: discrete valuation rings, Eisenstein polynomials, ramification, totally ramified extensions

6 comments

Today we consider the case of a totally ramified extension of local fields , with residue fields —recall that this means . It turns out that there is a similar characterization as for unramified extensions. (more…)

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

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

Tags: Chinese remainder theorem, Dedekind domains, ramification

1 comment so far

Tags: Chinese remainder theorem, Dedekind domains, ramification

1 comment so far

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 with quotient field ; let be a finite separable extension of , and the integral closure of in . We know that 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 , there is a prime lying above . 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 say . The primes contain and consequently lie above . Conversely, any prime of containing must lie above , since if is an ideal in a Dedekind domain contained in a prime ideal , then occurs in the prime factorization of (to see this, localize and work in a DVR). (more…)