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Definitions

An integer $n$ is represented by the binary quadratic form $ax^2 + bxy + cy^2$ if there exist integers $r$ and $s$ such that $n=ar^2 + brs + cs^2$

we call these two forms equivalent and, in general, binary quadratic form f is equivalent to $F(X, Y ) = f(\alpha X + \beta Y, \gamma X + \delta Y )$ whenever $ \alpha, \beta, \gamma, \delta$ are integers with $\alpha \delta - \beta \gamma = 1$ ,and so $f$ and $F$ represent the same integers. Therefore to determine what numbers are represented by a given binary quadratic form, we can study any binary quadratic form in the same equivalence class. If $f(x, y) = ax^2 + bxy + cy^2$ and $F(X, Y ) = AX^2 + BXY + CY^2$ above, note that $A = f(\alpha, \gamma), C = f(\beta, \delta)$ and $B^2 − 4AC = b^2 − 4ac$ $($in fact $B - b = 2(a \alpha \beta + b\beta \gamma + c\gamma \delta)$. For now I will focus on the case where the discriminant $d \ 0$ (since it is easier), $\\$ First note that if $f(x, y) = ax^2 + bxy + cy^2$ then $4af(x, y) = (2ax + by)^2 +|d|y^2$ and so is either always positive (if $a > 0$), else always negative. Replacing $f$ by $−f$ in the latter case we develop the theory of positive definite quadratic forms, and one can then easily deduce all the analogous results for negative definite $f$.

$\\$. The integers represented by the quadratic form gf(x, y) are of the form gn where n is represented by gf(x, y), so we assume that gcd(a, b, c) = 1. Gauss observed that it is possible to find a unique reduced binary quadratic form in each equivalence class, that is with $−a < b \leq a < c$ or $0 \leq b \leq a = c.$

Given any Binary Quadratic Form we can reduce it using Gauss Algorithm.

If $d \equiv 0 or 1 (mod 4)$ and is squarefree other than perhaps a factor of $4$ or $8$, then we say that $d$ is a fundamental discriminant. An algebraic integer is a number that is the root of a monic polynomial with integer coefficients; the algebraic integers in $ \mathbb{Q}(\sqrt{d})$ take the form $ \mathbb{Z}[\tau_d] := \{ m + n\tau_d : m, n ∈ \mathbb{Z} \} $, where $\tau_d= $\begin{cases} \text{$\frac{\sqrt{d}}{2}$,} &\quad\text{if $d \equiv 0 (mod 4)$}\\ \text{$\frac{1+\sqrt{d}}{2}$,} &\quad\text{if $d \equiv 1 (mod 4)$} \end{cases}

How do I establish theisomorphism between the equivalence classes of ideals of $\mathbb{Z}[\tau_d]$ presented in the form $\left(2a,b+\sqrt{d}\right)$ where $4a$ divides $b^2-d$, and binary quadratic forms of discriminant $d$

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