# References for Algebraic number theory

I am doing algebraic number theory first time. I have done all ring theory and field theory. I am interested in algebra , so also pretty much excited about algebraic number theory. I have a month's break before my semester begins. I want to self study some beginner's text covering my syllabus so that in class I can understand it much better, and may be it will turn up a research interest for me. Can somebody please suggest me some wonderful text for beginners in algebraic number theory, I do not want any number theory book using analytical methods. Strictly algebraic. Here are my contents...

Course Outline

Characteristic and minimal polynomial of an element relative to a finite extension, Equivalent definitions of norm and trace, Algebraic numbers, algebraic integers and their properties.

Integral bases, discriminant, Stickelberger’s theorem, Brille’s theorem, description of integral basis of quadratic, cyclotomic and special cubic fields.

Ideals in the ring of algebraic integers and their norm, factorization of ideals into prime ideals, generalised Fermat’s theorem and Euler’s theorem.

Dirichlet’s theorem on units, regulator of an algebraic number fields, explicit computation of fundamental units in real quadratic fields.

Dedekind’s theorem for decomposion of rational primes in algebraic number fields and its application, splitting of rational primes in quadratic and cyclotomic fields.

Any book that covers all these with best conceptual approach. Thanks!

Anyways here are some suggested readings by my institute-

Saban Alaca and Kenneth Williams, Introductory Algebraic Number Theory, Cambridge University Press (2003).

M. Ram Murty and J. Esmonde Problems in Algebraic Numbers Theory, Springer-Verlag (2004).

Erich Hecke, Lectures on the Theory of Algebraic Numbers, Springer-Verlag (1981).

Paula Ribenboim, Algebraic Numbers, John Wiley & Sons (1972).

Harry Pollard and Harold Diamond, The Theory of Algebraic Numbers, Dover Publications (2010).

Which one I should go for among them, or is there some other than these all, please suggest if any? Thanks!

• Dec 5, 2014 at 23:02
• Paula Ribenboim? Surely that Paulo? Anyway, I don't know if there's a single book that covers all the topics you want, But the Marcus Book is pretty good, as is Stewart and Tall. Dec 6, 2014 at 0:55
• Saban Alaca and Kenneth Williams Looks good to start with.. Dec 6, 2014 at 1:15
• I couldn't resist advertising Pierre Samuel's Algebraic Theory of Numbers. It's a very slim book so you can easily cover a lot in a month's time. Some of the proofs are really very elegant.
– SMG
Dec 15, 2014 at 1:32
• Does this answer your question? Beginner's text for Algebraic Number Theory Apr 24, 2021 at 23:06

Serge Lang's. Algebraic Number Theory is a good text also.

Here is a list of recommended texts from my institute K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd Edition, Springer-Verlag, Berlin, 1990.S. Lang, Algebraic Number Theory, Addison- Wesley, 1970.D. A. Marcus, Number Fields, Springer-Verlag, 1977.