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I learned and studied basic algebraic number theory (like number fields and extensions, prime decompositions, local fields, some of class field theory, ...) and I found that I'm not familiar with local fields yet. I know that global things are much harder than local things (because there are so many primes in global fields, but the local thing (local ring) has only one prime ideal which makes life easier) but I'm not good at treating things with local fields. Especially, I'll take an oral qual this semester and I want to find a good reference to review about local fields and related things (Hensel's lemma, local field extension, ramification, Krasner's lemma, ...). Here are some of the textbooks that I've read (not every single page)

  • Neukirch: This is the most famous algebraic number theory book which contains a lot of stuff including class field theory and Dedekind zeta functions. But I'm not sure if it is the best book to study the local theory.

  • Serre: Every Serre's book is very helpful but also very hard to read. His book 'Local Fields' looks like a dictionary, which is really hard to read carefully.

  • Milne: All of Milne's book is really kind and very easy to read. I prefer Milne's book, but the book is a little short and it may not be sufficient to study local theory.

I never read books like Cassels-Frolich or other classic books, so please recommend me if there are any good algebraic number theory books to read and review all the basic things.

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May I mention my own book, A gentle course in local class field theory ? It just appeared.

--Pierre Guillot

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  • $\begingroup$ I read a few of your books too, and it is a good source for the topic. Thank you for a good book! $\endgroup$ – Seewoo Lee Jan 24 at 4:42
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My favorite is Cassels' Local Fields. Turn to Serre afterwards.

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  • $\begingroup$ I'm reading it now and it seems good! $\endgroup$ – Seewoo Lee Jan 21 at 5:51

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