Author: Wojowu (Page 2 of 2)

Decomposition and inertia field degrees

We shall establish relations between degrees, inertia degrees and ramification indices involved in decomposition and inertia field of a given prime in Galois extension. This is based on Marcus’s Number Fields and online notes by R. Ash. It follows a very similar appoach to, but is not based on, the one which can be found in this blog post by Sander Mack-Crane. For this blog post, understanding of Galois theory and basic facts about number fields is necessary.

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Invertibility of ideals in Dedekind domains

Based on arguments in Marcus’s Number Fields and K. Conrad’s expository paper on ideal factorization. I assume familiarity with the concepts related to ideals and fractional ideals in a commutative ring.

Recall that an integral domain \(R\) with the field of fractions \(K\) is called a Dedekind domain if the following conditions hold:

  • \(R\) is Noetherian, so every nonempty set of ideals has a maximal element, or equivalently, every ideal is finitely generated,
  • every prime ideal in \(R\) is a maximal ideal, and
  • \(R\) is integrally closed in \(K\), so that every root of a monic polynomial from \(R[x]\) lying in \(K\) lies in \(R\).

Our goal is to prove the following proposition:

Proposition: Assume \(R\) is a Dedekind domain. For any nonzero (i.e. containing a nonzero element) fractional ideal \(I\) in \(K\) there exists a fractional ideal \(J\) such that \(IJ=R\).

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Restarting the blog

Alright, it’s been almost 4 years since I have created this blog and I still haven’t posted anything apart from the introduction post. This wasn’t caused by the lack of time or anything like that, but rather by me being lazy and not motivated. But today I’ve decided to get around and get something done on this blog.

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Hi, my name is Wojtek, I am middle school student in Poland. I created this blog inspired by Adam  P. Goucher’s blog – Complex Projective 4-space. Like in his blog, I want to write about math stuff. I won’t write here about basics, so I don’t mind if you don’t understand everything 🙂

Because of personal difficulties, Posts will appear very irregularly, but I’ll do my best. Hope to see any feedback soon.

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