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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.

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    $\begingroup$ 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. $\endgroup$ Oct 30, 2014 at 18:02
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    $\begingroup$ 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 :) $\endgroup$
    – Prism
    Nov 23, 2014 at 10:55
  • $\begingroup$ Have you had a look at any of the books recommended? Anything to report back? $\endgroup$ Dec 13, 2014 at 8:24

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

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

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

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