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I am not so familiar with the theory of measures which Andre Weil uses to develope the Class Field Theory.
However, I am interested in learning algebraic number theory and I recently found that the basic ideas of commutative algebra are not so familiar with me:).
Besides, the fundamental notion of algebraic number theory in Neukirch's book Algebraic number theory is the Minkowski Theory which is quite unclear to me.
Is there any book except for those of Bourbaki on the topic of commutative algebra and Minkowski theory such that it is friendly to beginners?
Thank you very much.

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  • $\begingroup$ By Minkowski Theory, do you mean geometry of numbers / convex body theorem stuff, or something else? $\endgroup$ Feb 9, 2011 at 15:11
  • $\begingroup$ Yes, it is exactly that what you mean. $\endgroup$
    – awllower
    Feb 10, 2011 at 1:15
  • $\begingroup$ Although there is another subject also called geometry of numbers and is essentially different from Minkowski Theory which is why I took this name following Neukirch. $\endgroup$
    – awllower
    Feb 10, 2011 at 1:19
  • $\begingroup$ Weil worked on conformal field theory?? I never knew! (I kid. But seriously: down with unexplained abbreviations!) $\endgroup$ Feb 10, 2011 at 9:02
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    $\begingroup$ You use the abbreviation "CFT", which among other things could stand for Conformal Field Theory or Class Field Theory. ("Field" means different things in the two contexts, for what it's worth.) I assume you mean class field theory. I was trying to point out, gently, that using abbreviations like this without defining them is almost never a good idea. The time it takes you to type it out in full will save many other people from having to spend time figuring out what you mean. It's a courtesy, in other words. $\endgroup$ Feb 10, 2011 at 14:46

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You might try Pierre Samuel, "Algebraic Number Theory", for a concise introduction with basic treatments of what you are asking about.

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  • $\begingroup$ Well, then what about the book Commutative Algebra by Pierre Samuel? $\endgroup$
    – awllower
    Feb 11, 2011 at 3:38
  • $\begingroup$ If you mean Zariski-Samuel, it is oriented towards the needs of algebraic geometers. $\endgroup$ Feb 11, 2011 at 7:02
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    $\begingroup$ Re: Zariski and Samuel: it is two volumes. The first volume remains one of the more graceful, lucid takes on commutative algebra. It is the second volume which tackles more specialized material in the service of (not exactly au courant but still of some interest) algebraic geometry. So it was quite maddening to me when I recently tried to order both volumes and found that the first volume (only) was out of print. It's this kind of thing that makes me want to kiss all publishers (where of course I don't mean kiss, but I don't really mean the other word either so I won't write it). $\endgroup$ Feb 11, 2011 at 13:22
  • $\begingroup$ As an introductory book on commutative algebra Atiyah and MacDonald has the advantage (huge, in my eyes) of being short. It is also very clear. $\endgroup$
    – inkspot
    Mar 2, 2011 at 20:01

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