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I don't know if this already has an answer here, however I checked for similar questions and could find none.

I am an undergraduate student in Mathematics. For my summer project, my supervisor advised me to study Local Fields following 'J.P. Serre's Local Fields'. I had not had any course in Galois Theory, Commutative Algebra, etc. till now ( I have these courses in my upcoming semester). As I went through the book, I saw a lot of commutative algebra at play. Similarily field theory and galois theory are also needed. Upon discussing this problem of prerequisites with my supervisor, he advised me to just look up the definitions and move on. He also suggests "you need not get full thorough understanding of this at this first encounter", and that "Maths is to be read non-linearly". But lately, I am having problems complying to his philosophy. I am getting lost in working through small details in the book, which the book just mentions as some fact/trivial statement, and I think this is precisely because I don't have enough familiarity with the objects that I am working with. Due to this, I am not getting any feel of the subject whatsoever. Moreover, at each line/step, I have to look up definitions which disturbs the flow of reading. So, I think it is better for me to first go through the prerequisites of the book. Hence, I would like to know :

What minimum level of prerequisite do I need to sail through this book? I know that even after being familiar with these prerequisites, I might run into looking some definition/concept, but atleast the condition would be better than what it is as of now. You may suggest books that go through the same subject at some basic level. I would appreciate any kind of help.

P-addict
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    In my opinion, your advisor is sending you into an impossible task. A sure formula for crash-and-burn. You don't have the prerequisites, and just looking up things as you go is not sufficient. I would drop this assigment, else your confidence will be severely impacted. – quasi Jul 05 '19 at 07:23
  • thanks, I share the same feeling. I am going to change the topic of the project to some topic/book that is needed for reading this book. So, it would really help if you can give me some advise on such topics/books. – P-addict Jul 05 '19 at 07:25
  • I believe it is definitely useful (and possible in a reasonable amount of time) to study some basic Galois theory first. Any Algebra book should cover the essentials, I would suggest you look for a rather concise treatment of the subject in order not to lose too much time. – asdq Jul 05 '19 at 07:26
  • Sure. I was looking for reading Artin's Galois Theory. Is it a good choice? I would appreciate other suggestions. – P-addict Jul 05 '19 at 07:27
  • @P-addict I completely agree with quasi, and I would advise to take Group Theory/Ring Theory (algebraic Structures I in my university), and then Fields Theory with Galois Theory (=Algebraic Structures II), and after this a basic in Algebraic Numbers Theory. Highly recommended to study also some commutative algebra, if only to widen up your views. – DonAntonio Jul 05 '19 at 07:28
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    I can recommend Lang's Algebra book. – asdq Jul 05 '19 at 07:30
  • @DonAntonio Thanks a lot. I already have taken Group Theory and RIng Theory. I am looking forward to going through Field Theory/Galois theory and commutative algebra. I believe even Algebraic Number Theory requires a bit of commutative algebra, so I thought I would go for it later. – P-addict Jul 05 '19 at 07:30
  • In Algebra, I had Group Theory, Rings, Modules, Vector Spaces, Linear Alegbra .How good is "Number Fields by Marcus" for me given my background in Algebra? – P-addict Jul 05 '19 at 07:35
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    @P-addict: For Galois Theory, I enjoyed the text: Kaplansky -- Fields and Rings, 2nd Ed (1972), pages 1-77. Brief, elegant treatment, and prerequisites match what you have. – quasi Jul 05 '19 at 07:35
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    The Book by Artin (Emil) is nice...but a little too old already (from 1942, with new editions in the 60's and 70's of the past century). The book "Algebra" by Artin (Michael...another, newer one :) ) can be more fit to do a quicky on Galois Theory. There are many books on the subject: Lang's, Reid's, Milne's, etc., but this theme may be also included in many good Algebra books, like Dummit&Foote's, Hungerford's, Rotman's, etc. I'd recommend you go to your school's Mathematics Library and invest some time reading the books there and choosing from then. – DonAntonio Jul 05 '19 at 07:35
  • @DonAntonio :), thanks for the suggestions. Turns out this "another, newer one" is actually a kid, or better, son, of the old one :) – P-addict Jul 05 '19 at 07:38
  • @P-Addict: As regards the text "Marcus -- Number Fields, 2nd Ed (2018)", in the preface, the authors states "The level of exposition rises as the book progresses. In Chap. 2, for example, the degree of a field extension is defined, while in Chap. 4 it is assumed that the reader knows Galois theory.". Based on that, you should be OK for the first 3 chapters (which is probably just fine for a summer project). – quasi Jul 05 '19 at 07:47
  • @P-addict Yes, I know...hehe. And I think he'd thank very much the title "kid", as he's already 84/85 ... – DonAntonio Jul 05 '19 at 07:48

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