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I do not see what Local fields they are Talking about in the first chapter of Algebraic Number theory book by Cassels, Frohlich.

Index of first chapter is

  1. Discrete Valuation Rings
  2. Dedekind Domains
  3. Modules and Bilinear Forms
  4. Extensions
  5. Ramification
  6. Totally Ramified Extensions
  7. Non-ramified Extensions
  8. Tamely Ramified Extensions
  9. The Ramification Groups
  10. Decomposition

I did not read line by line but I think I looked at that carefully.

I do not see any thing related to Local fields. But title is Local fields.

Some classification :

Every local field is isomorphic (as a topological field) to one of the following:

Archimedean local fields (characteristic zero): the real numbers $\mathbb{R}$, and the complex numbers $\mathbb{C}$.

Non-archimedean local fields of characteristic zero: finite extensions of the p-adic numbers $\mathbb{Q}_p$ (where $p$ is any prime number).

Non-archimedean local fields of characteristic $p$ (for $p$ any given prime number): finite extensions of the field of formal Laurent series $\mathbb{F}_q((T))$ over a finite field $\mathbb{F}_q$ (where $q$ is a power of $p$).

So, I want to know what did I miss.

What is the reason behind the title. Or, did I not read carefully?

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closed as unclear what you're asking by Adam Hughes, Alex Mathers, Daniel W. Farlow, Leucippus, user91500 Sep 22 '16 at 6:15

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    $\begingroup$ Did you read the introduction? $\endgroup$ – quid Sep 20 '16 at 19:55
  • $\begingroup$ @quid I did not read the introduction.. I. Will read now $\endgroup$ – user87543 Sep 20 '16 at 20:17
  • $\begingroup$ Alright. The reason I ask is that the question is specifically addressed there. The short is, do not make too much of the title. $\endgroup$ – quid Sep 20 '16 at 20:22
  • $\begingroup$ @quid there is nothing mentioned about it in introduction... $\endgroup$ – user87543 Sep 20 '16 at 20:23
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    $\begingroup$ It is the second paragraph where it says it could also be called "Algebraic theory of Dedekind domains." $\endgroup$ – quid Sep 20 '16 at 20:26