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I have experience in abstract algebra up to fields and field extensions using Artin's Algebra. I am wondering what book would be the most user friendly but also rigorous introduction to algebraic number theory.

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    $\begingroup$ Consider Number Fields by Marcus $\endgroup$ May 8, 2020 at 23:25
  • $\begingroup$ I like Fermat's Last Theorem by Harold M. Edwards. I'm writing this as a comment because I don't know much about ANT. $\endgroup$
    – Jack M
    May 9, 2020 at 0:55
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    $\begingroup$ I second Marcus. Edwards' book is a good source for historians, but not for introducing yourself to algebraic number theory. $\endgroup$ May 9, 2020 at 8:42
  • $\begingroup$ See also mathoverflow.net/questions/13282/… $\endgroup$
    – J W
    Sep 10, 2021 at 9:58

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I would recommend Number Fields by Marcus. It is described on that linked web site as follows:

  • Contains over 300 exercises

  • Assumes only basic abstract algebra

  • Covers topics leading up to class field theory

Recently a second edition was published; that is reviewed here.

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    $\begingroup$ I was wondering when someone would produce an edition without horrible typesetting! Thanks for the update. $\endgroup$
    – Favst
    May 11, 2020 at 13:29
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    $\begingroup$ @Favst The typesetting is nice, but my copy began to fall apart after a day or so's use, and has been saved from further deterioration only by a great deal of sticky tape. $\endgroup$ May 11, 2020 at 15:44
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Algebraic Number Theory and Fermat's Last Theorem by Stewart and Tall is an amazing introduction into the various modern topics. It revisits the basic concepts of rings and fields and then dives into advanced topics such as class field theory.

For more information, see:

https://www.maa.org/press/maa-reviews/algebraic-number-theory-and-fermats-last-theorem-0

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    $\begingroup$ It's the first thing that shows up on a Google search $\endgroup$ May 9, 2020 at 0:39
  • $\begingroup$ @MichaelMorrow My browser marks it as untrusted due to an expired SSL certificate. I’m surprised that Google is ranking it highly: which search term did you use? $\endgroup$
    – Erick Wong
    May 10, 2020 at 2:17
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    $\begingroup$ To be on the safe side, I have edited the post. $\endgroup$ May 10, 2020 at 2:53
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    $\begingroup$ @MichaelMorrow Thank you for the edit! I have flipped my downvote (which was purely in the interest of decreasing the rank of the answer in case it was truly unsafe) into an upvote (it’s far more useful to not have the reference buried behind a link, and now that I can see it I agree with the recommendation). $\endgroup$
    – Erick Wong
    May 10, 2020 at 20:16
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Algebraic Number Theory by Jarvis has the advantage (for self-study) of having extensive hints and solutions to exercises in an appendix. It's also very accessible - a light knowledge of groups, rings and fields, along with some elementary number theory and linear algebra is plenty. Topics such as ideals and quotient rings are introduced in the text. It's gentler than Marcus or Stewart & Tall, which both require a stronger background in abstract algebra.

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