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Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.

From experiments with the website http://www.numbertheory.org/php/classnopos.html and comparison with my own programs, it appeared that the division in half amounted to identifying the distinct forms (whenever the principal form does not represent $-1$) $$ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{with} \; \; \langle -\alpha, \beta, \gamma \rangle, $$ where these are ``reduced'' in the sense of Gauss and Lagrange when $$ \alpha \gamma > 0 \; \; \mbox{and} \; \; \beta > |\alpha - \gamma|. $$

So it is needed to show that these really are distinct classes when $1$ and $-1$ are distinct as forms. However, this is not hard. If the two forms above are equivalent, then the opposite of $\langle \alpha, \beta, -\gamma \rangle$ in the form class group is $ \langle \gamma, \beta, -\alpha \rangle. $ Thus the hypothesis amounts to $$ 1 =\langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle. $$ We are allowed to insist that $\gcd(\alpha,\beta ) = 1,$ important. Can always be arranged, although the result may not be ``reduced'' any longer.

The algorithm of Shanks, Buell Binary Quadratic Forms, pages 64-65, tells us that, with $\gcd(\alpha,\beta ) = 1,$ $$ \langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle = \langle \alpha \gamma, \beta, -1 \rangle. $$ In short, the hypothesis that $ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{and} \; \; \langle -\alpha, \beta, \gamma \rangle $ are equivalent implies directly that the principal form represents $-1.$ Very satisfying from my point of view. Note that we do not need to use Shanks, various books discuss the ``united forms'' approach of Dirichlet, pages 55-57 in Buell. On page 57, he confirms, with $\color{green}{\gcd(a_1,a_2,B)= 1},$ that $$\color{magenta}{ \langle a_1, B, a_2C \rangle \circ \langle a_2, B, a_1C \rangle = \langle a_1 a_2,B, C \rangle}. $$ Dirichlet gives the same outcome as the Shanks method, but with no additional $\gcd$ assumptions, using $$\color{magenta}{ a_1 = \alpha, a_2 = \gamma, B = \beta, C = -1.} $$ This is also Theorem 98 on page 138 of Leonard Eugene Dickson, Introduction to the Theory of Numbers.

Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.

From experiments with the website http://www.numbertheory.org/php/classnopos.html and comparison with my own programs, it appeared that the division in half amounted to identifying the distinct forms (whenever the principal form does not represent $-1$) $$ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{with} \; \; \langle -\alpha, \beta, \gamma \rangle, $$ where these are ``reduced'' in the sense of Gauss and Lagrange when $$ \alpha \gamma > 0 \; \; \mbox{and} \; \; \beta > |\alpha - \gamma|. $$

So it is needed to show that these really are distinct classes when $1$ and $-1$ are distinct as forms. However, this is not hard. If the two forms above are equivalent, then the opposite of $\langle \alpha, \beta, -\gamma \rangle$ in the form class group is $ \langle \gamma, \beta, -\alpha \rangle. $ Thus the hypothesis amounts to $$ 1 =\langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle. $$ We are allowed to insist that $\gcd(\alpha,\beta ) = 1,$ important. Can always be arranged, although the result may not be ``reduced'' any longer.

The algorithm of Shanks, Buell Binary Quadratic Forms, pages 64-65, tells us that, with $\gcd(\alpha,\beta ) = 1,$ $$ \langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle = \langle \alpha \gamma, \beta, -1 \rangle. $$ In short, the hypothesis that $ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{and} \; \; \langle -\alpha, \beta, \gamma \rangle $ are equivalent implies directly that the principal form represents $-1.$ Very satisfying from my point of view. Note that we do not need to use Shanks, various books discuss the ``united forms'' approach of Dirichlet, pages 55-57 in Buell. On page 57, he confirms, with $\color{green}{\gcd(a_1,a_2,B)= 1},$ that $$ \langle a_1, B, a_2C \rangle \circ \langle a_2, B, a_1C \rangle = \langle a_1 a_2,B, C \rangle. $$ Dirichlet gives the same outcome as the Shanks method, but with no additional $\gcd$ assumptions, using $$ a_1 = \alpha, a_2 = \gamma, B = \beta, C = -1. $$ This is also Theorem 98 on page 138 of Leonard Eugene Dickson, Introduction to the Theory of Numbers.

Put another way, when the principal form does not represent $1,$ we get a distinct form that does represent $-1,$ and Dirichlet says $$\color{magenta}{ \langle \alpha, \beta, -\gamma \rangle \circ \langle -1, \beta, \alpha \gamma\rangle = \langle -1, \beta, \alpha \gamma\rangle \circ \langle \alpha, \beta, -\gamma \rangle = \langle -\alpha , \beta, \gamma \rangle}. $$ As $\langle -1, \beta, \alpha \gamma\rangle$ is not the principal class, the result of the Gauss composition gives a different class from the original, therefore $\langle \alpha, \beta, -\gamma \rangle$ and $\langle -\alpha , \beta, \gamma \rangle$ must be distinct.

Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.

From experiments with the website http://www.numbertheory.org/php/classnopos.html and comparison with my own programs, it appeared that the division in half amounted to identifying the distinct forms (whenever the principal form does not represent $-1$) $$ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{with} \; \; \langle -\alpha, \beta, \gamma \rangle, $$ where these are ``reduced'' in the sense of Gauss and Lagrange when $$ \alpha \gamma > 0 \; \; \mbox{and} \; \; \beta > |\alpha - \gamma|. $$

So it is needed to show that these really are distinct classes when $1$ and $-1$ are distinct as forms. However, this is not hard. If the two forms above are equivalent, then the opposite of $\langle \alpha, \beta, -\gamma \rangle$ in the form class group is $ \langle \gamma, \beta, -\alpha \rangle. $ Thus the hypothesis amounts to $$ 1 =\langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle. $$ We are allowed to insist that $\gcd(\alpha,\beta ) = 1,$ important. Can always be arranged, although the result may not be ``reduced'' any longer.

The algorithm of Shanks, Buell pages 64-65, tells us that, with $\gcd(\alpha,\beta ) = 1,$ $$ \langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle = \langle \alpha \gamma, \beta, -1 \rangle. $$ In short, the hypothesis that $ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{and} \; \; \langle -\alpha, \beta, \gamma \rangle $ are equivalent implies directly that the principal form represents $-1.$ Very satisfying from my point of view.

Necessary detail for when $\gcd(\alpha,\beta ) \neq 1;$ if $\langle A,B,C \rangle \equiv \langle P,Q,R \rangle$ by $SL_2(\mathbb Z)$ matrix $$ \left( \begin{array}{rr} S & T \\ U & V \end{array} \right) , $$ then $\langle -A,B,-C \rangle \equiv \langle -P,Q,-R \rangle$ by $SL_2(\mathbb Z)$ matrix $$ \left( \begin{array}{rr} -S & T \\ U & -V \end{array} \right). $$ In particular, taking the inverse and negating throughout, $\langle -P,Q,-R \rangle \equiv \langle -A,B,-C \rangle$ by $SL_2(\mathbb Z)$ matrix $$ \left( \begin{array}{rr} V & T \\ U & S \end{array} \right). $$ What I mean by this is, given Gram (or Hessian) matrix $G$ for $\langle A,B,C \rangle $ and the indicated 2 by 2 matrix, call it W, we get the Gram matrix for the new form by $W^T G W.$

Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.

From experiments with the website http://www.numbertheory.org/php/classnopos.html and comparison with my own programs, it appeared that the division in half amounted to identifying the distinct forms (whenever the principal form does not represent $-1$) $$ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{with} \; \; \langle -\alpha, \beta, \gamma \rangle, $$ where these are ``reduced'' in the sense of Gauss and Lagrange when $$ \alpha \gamma > 0 \; \; \mbox{and} \; \; \beta > |\alpha - \gamma|. $$

So it is needed to show that these really are distinct classes when $1$ and $-1$ are distinct as forms. However, this is not hard. If the two forms above are equivalent, then the opposite of $\langle \alpha, \beta, -\gamma \rangle$ in the form class group is $ \langle \gamma, \beta, -\alpha \rangle. $ Thus the hypothesis amounts to $$ 1 =\langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle. $$ We are allowed to insist that $\gcd(\alpha,\beta ) = 1,$ important. Can always be arranged, although the result may not be ``reduced'' any longer.

The algorithm of Shanks, Buell Binary Quadratic Forms, pages 64-65, tells us that, with $\gcd(\alpha,\beta ) = 1,$ $$ \langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle = \langle \alpha \gamma, \beta, -1 \rangle. $$ In short, the hypothesis that $ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{and} \; \; \langle -\alpha, \beta, \gamma \rangle $ are equivalent implies directly that the principal form represents $-1.$ Very satisfying from my point of view. Note that we do not need to use Shanks, various books discuss the ``united forms'' approach of Dirichlet, pages 55-57 in Buell. On page 57, he confirms, with $\color{green}{\gcd(a_1,a_2,B)= 1},$ that $$\color{magenta}{ \langle a_1, B, a_2C \rangle \circ \langle a_2, B, a_1C \rangle = \langle a_1 a_2,B, C \rangle}. $$ Dirichlet gives the same outcome as the Shanks method, but with no additional $\gcd$ assumptions, using $$\color{magenta}{ a_1 = \alpha, a_2 = \gamma, B = \beta, C = -1.} $$ This is also Theorem 98 on page 138 of Leonard Eugene Dickson, Introduction to the Theory of Numbers.

Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.

From experiments with the website http://www.numbertheory.org/php/classnopos.html and comparison with my own programs, it appeared that the division in half amounted to identifying the distinct forms (whenever the principal form does not represent $-1$) $$ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{with} \; \; \langle -\alpha, \beta, \gamma \rangle, $$ where these are ``reduced'' in the sense of Gauss and Lagrange when $$ \alpha \gamma > 0 \; \; \mbox{and} \; \; \beta > |\alpha - \gamma|. $$

So it is needed to show that these really are distinct classes when $1$ and $-1$ are distinct as forms. However, this is not hard. If the two forms above are equivalent, then the opposite of $\langle \alpha, \beta, -\gamma \rangle$ in the form class group is $ \langle \gamma, \beta, -\alpha \rangle. $ Thus the hypothesis amounts to $$ 1 =\langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle. $$ We are allowed to insist that $\gcd(\alpha,\beta ) = 1,$ important. Can always be arranged, although the result may not be ``reduced'' any longer.

The algorithm of Shanks, Buell pages 64-65, tells us that, with $\gcd(\alpha,\beta ) = 1,$ $$ \langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle = \langle \alpha \gamma, \beta, -1 \rangle. $$ In short, the hypothesis that $ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{and} \; \; \langle -\alpha, \beta, \gamma \rangle $ are equivalent implies directly that the principal form represents $-1.$ Very satisfying from my point of view.

Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.

From experiments with the website http://www.numbertheory.org/php/classnopos.html and comparison with my own programs, it appeared that the division in half amounted to identifying the distinct forms (whenever the principal form does not represent $-1$) $$ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{with} \; \; \langle -\alpha, \beta, \gamma \rangle, $$ where these are ``reduced'' in the sense of Gauss and Lagrange when $$ \alpha \gamma > 0 \; \; \mbox{and} \; \; \beta > |\alpha - \gamma|. $$

So it is needed to show that these really are distinct classes when $1$ and $-1$ are distinct as forms. However, this is not hard. If the two forms above are equivalent, then the opposite of $\langle \alpha, \beta, -\gamma \rangle$ in the form class group is $ \langle \gamma, \beta, -\alpha \rangle. $ Thus the hypothesis amounts to $$ 1 =\langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle. $$ We are allowed to insist that $\gcd(\alpha,\beta ) = 1,$ important. Can always be arranged, although the result may not be ``reduced'' any longer.

The algorithm of Shanks, Buell pages 64-65, tells us that, with $\gcd(\alpha,\beta ) = 1,$ $$ \langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle = \langle \alpha \gamma, \beta, -1 \rangle. $$ In short, the hypothesis that $ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{and} \; \; \langle -\alpha, \beta, \gamma \rangle $ are equivalent implies directly that the principal form represents $-1.$ Very satisfying from my point of view.

Necessary detail for when $\gcd(\alpha,\beta ) \neq 1;$ if $\langle A,B,C \rangle \equiv \langle P,Q,R \rangle$ by $SL_2(\mathbb Z)$ matrix $$ \left( \begin{array}{rr} S & T \\ U & V \end{array} \right) , $$ then $\langle -A,B,-C \rangle \equiv \langle -P,Q,-R \rangle$ by $SL_2(\mathbb Z)$ matrix $$ \left( \begin{array}{rr} -S & T \\ U & -V \end{array} \right). $$ In particular, taking the inverse and negating throughout, $\langle -P,Q,-R \rangle \equiv \langle -A,B,-C \rangle$ by $SL_2(\mathbb Z)$ matrix $$ \left( \begin{array}{rr} V & T \\ U & S \end{array} \right). $$ What I mean by this is, given Gram (or Hessian) matrix $G$ for $\langle A,B,C \rangle $ and the indicated 2 by 2 matrix, call it W, we get the Gram matrix for the new form by $W^T G W.$