# Leads for penetrating the field of Algebraic number theory

I need to rapidly get up to speed on the following topics, for the purposes of an internship:

1. Global and local fields.
2. Localization.
3. Number fields, function fields, etc.
4. Ring of integers, field of fractions.
5. The property of a ring $R$ that "for any non-zero ideal $I$ of $R$, $R/I$ is finite".

These are some of the keywords and topics that my professor brought up in a short crash course on the topics I need to read up about this week. I borrowed Serge Lang's "Algebraic number theory" but I've found it too formal and abstract, I'm hoping for more motivational books. I also may need something slightly more elementary than Lang, since I haven't covered module theory and am a shaky on everything except the most elementary ring theory (the topics I've mostly covered in class are the isomorphism theorems, the various kinds of ideals, and the implications between euclidean, integral, principal etc).

I've found it difficult to really get started. Can anyone suggest a path I could follow in my reading, and ideally some book titles?

• "Only" that ? Hopefully you already had basic courses in linear algebra, group and ring theory, some commutative algebra...? Point number 5 is odd: it isn't true in the general case, of course, though in some cases it is, yet I cannot seem to locate it safely within the other subjects you mention. – Timbuc Jan 21 '15 at 18:02
• @Timbuc Point 5 is a property which supposedly characterizes a useful class of rings. I have of course also studied group and field theory, and linear algebra. – Jack M Jan 21 '15 at 18:08
• I know that, Those rings are sometimes called "residually finite", in similarity with groups (though in group it means another thing) – Timbuc Jan 21 '15 at 18:12
• There are some great books in the subject and you will have to check them in order to choose the one that fits you better. First, Global and Local Fields is advanced stuff you don't need for an introduction to algebraic number theory. Second, here some authors whose books you should check: Janusz , Frohlich-Taylor, Neukirch, Cassels-Frohlic, Ribenboim, Narkiewicz, Cohn...and many other more. – Timbuc Jan 21 '15 at 18:16