Totally ramified extensionsOctober 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…)

e, f, and the remainder theoremSeptember 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…)