## Abstraction

The book “Number Fields” by D. Marcus is a very well-known introductory book on algebraic number theory. Its most memorable aspect is, without a doubt, the great number of exercises it contains. They vary from short(ish) computational exercises, through various technical results used later in the book, to series of exercises aimed to establish (sometimes very deep) results in number theory. They are structured in a way which allows even an unexperienced reader be able to solve most, if not all, exercises even on their first reading, thanks to (often very elaborate) hints provided.

However, even then a reader might want to refer some external source in order to see how the exercise can be solved, because otherwise it might be difficult to proceed any further (I myself would appreciate such a source at times). And, as they say, if you want something done right, do that yourself.

I’ve been thinking about this project for a short while already, and recently I have finally decided to start working on it. At the time of publishing this post I have finished writing up solutions to exercises from chapter 1. More info, including a link to the actual file, can be found here (a link to that page can be also found on the sidebar). Please put all feedback under that page. With each further chapter completed that page will be completed, and I don’t plan on making posts like this one until the project is completed.

Since the post title promised some info, I’d like to mention that for three reasons the amount of content appearing on the blog in the near future will not be as large as it was over past two weeks (I am not putting this on hiatus though). First is this project, since I want to have it done at some point in the future, which means I will have to invest some time into it. Second is a trip I am going to next week. I might or might not work on some post while I’m there, we will see. Third, university starts at the beginning of October, but I still should be able to work on the blog, at least during weekends, but most likely also during the week.

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1. #### sylvainjulien

You seem to focus on algebraic number theory. Do you (plan to) specialize in this field (no pun intended)?

• #### Wojowu

Recently ANT has been what I have been learning about, since it’s a field I wanted to get to know for a while, and it just so happened that the idea of reviving the blog came to me around now. I don’t have a clearly chosen field I want to specialize in, but for now I want to learn more ANT.

• #### sylvainjulien

I see. Do you think ANT is required to study the structure of general classes of L-functions or would a mix of analytic number theory and algebra be enough? I was surprised, several years ago, to not get any result for “automorphisms of the Selberg class” in Google and had no choice but trying to define them myself. I don’t think my mathematical background is sufficient for that purpose though.

• #### Wojowu

I would guess that at least some ANT would be necessary, or at least somewhat useful, to know. I don’t know what the theory of “general classes of L-functions” exactly deals with, but Google search told me that some large classes of L-functions come from ANT or class field theory considerations (like Artin or Hecke L-functions). I would say I am a wrong person to ask this kind of question.