Tagged Questions

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Algorithm for determining whether two real quadratic numbers are equivalent under a modular transformation

Let $\alpha \in \mathbb{C}$ be an algebraic number. If the minimal plynomial of $\alpha$ over $\mathbb{Q}$ has degree $2$, we say $\alpha$ is a quadratic number. Then $\alpha$ is a root of a unique ...
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Binary quadratic forms whose discriminant is that of a quadratic number field

Let $K$ be a quadratic number field, $D$ its discriminant. Let $ax^2 + bxy + cy^2$ be an integral binary quadratic form such that $D = b^2 - 4ac$. It seems that gcd$(a, b, c) = 1$(see this question). ...
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68 views

How to compute the class group of an order of a quadratic number field

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, i.e. the subring of $K$ which is a free $\mathbb{Z}$-module of rank $2$.Let $D$ be its discriminant. We use the notation and the result ...
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The inverse class of the class represented by a primitive binary quadratic form of discriminant $D$

We use the definitions of this question. 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 $D \equiv 1$ ...
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space of “shapes” of rings of integers of number fields

In the recent paper by Bhargava and Harron, arXiv:1309.2025. They prove that cubic, quartic and quintic fields have equidistributed "shapes" as discriminant gets large. Their notion of shape ...
There is a group structure of binary quadratic forms of given discriminant $d$: Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$ The composition of two binary quadratic ...
A characterization of an ambiguous class of binary quadratic forms of discriminant $D$
We use the definitions of this question. Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). There exists a bijection $\psi\colon Cl^+(R) \rightarrow C(D)$ by ...