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Let $F = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $F$. If $D$ is not a square integer and gcd($a, b, c) = 1$, we say $ax^2 + bxy + cy^2$ is primitive.

Is the following proposition true? If yes, how do we prove it?

Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). By this question, there exists an order $R$ of a quadratic number field $K$ such that the discriminant of $R$ is $D$. Let $ax^2 + bxy + cy^2$ be a binary quadratic form whose discriminant is $D$. Then $I = \mathbb{Z}a + \mathbb{Z}\frac{(-b + \sqrt{D})}{2}$ is an ideal of $R$. Moreover, $I$ is invertible if and only if $ax^2 + bxy + cy^2$ is primitive.

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    Some off-topic discussion of meta-matters (downvoting, etc) has been purged. Please refrain from discussing meta matters on the main site. Instead, please use the meta site.2012-10-16
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    How is this elementary-number-theory? Per chance a retag is in order?2013-02-22
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    @awllower The theory of integral binary quadratic forms belongs to elementary number theory.2013-02-22
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    @MakotoKato The introduction to this tag does not specify the point, and I think this question, quite involved with algebraic notions such as number fields, or ideals..., should pertain to algebraic number theory. Of course, this could be said to be a problem of taste, while tastes differ.2013-02-23
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    @awllower I have a book titled "elementary number theory" written by Teiji Takagi in Japanese. The author is one of the main pioneers of class field theory. The book uses only elementary methods as the title shows. It contains the theory of integral binary quadratic forms and the theory of quadratic number fields. I would like to point out that quadratic number fields can be treated quite elementarily without the knowledge of algebraic number theory. The book, however, does not provide an answer to my question.2013-02-23
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    @awllower I believe the problem can be solved without the deep knowledge of algebraic number theory. For example, I don't think we need the fact that the ring of integers of a quadratic number field is a Dedekind domain.2013-02-23
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    @awllower The question asks the relation between the theory of integral binary quadratic forms and the theory of quadratic number fields. As I said, the former belongs to elementary number theory, while the latter belongs to algebraic number theory. In short, I think there is no clear cut tags.2013-02-23
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    @MakotoKato Thanks for indicating this relation. I forgot that we can treat binary quadratic form in purely elementary methods. Indeed, there should not be clear-cut tags. Thanks for reminding me of these facts.2013-02-23

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