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Could anyone direct me to a good reference book(s) for quadratic integer rings? Ideally, the reference would begin with their elementary properties and then proceed through their ring-theoretic properties: for example, which quadratic integer rings are PIDs, which are UFDs, which are EDs, and which are multi-stage EDs. Also, if the reference could connect the subject material to elementary number theory that would be splendid. For example, connecting the primes of $\Bbb Z[i]$ with the primes of $\Bbb Z$ and solutions to Pell's Equation.

I've only read about these rings through books whose main purpose was to introduce the the fundamentals of abstract algebra, and I want a more specialized reference.

Any input is appreciated.

learning_math
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    I don't know if there's a ton on specifically quadratic rings, but I know that Marcus' Number Fields addresses them specifically and explicitly. – Adam Hughes Oct 30 '14 at 18:02
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    I am currently reading "Algebraic Number Theory" by Frazer Jarvis (from Springer Undergraduate Texts series), and it is very good. The chapter 6 is titled "Imaginary Quadratic Fields", but since I am not there yet I cannot say anything more :) – Prism Nov 23 '14 at 10:55
  • Have you had a look at any of the books recommended? Anything to report back? – Gerry Myerson Dec 13 '14 at 08:24

3 Answers3

3

Pierre Samuel's "Algebraic Theory of Numbers" is a standard reference for this kind of questions.

Olórin
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Stewart and Tall, Algebraic Number Theory, doesn't do everything you want, but does quite a lot of it.

Gerry Myerson
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Ireland and Rosen's book A classical Introduction to Modern Number Theory

P Vanchinathan
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