Exercise 3.4.8 (Discriminant of a product of linear forms)

Question

Show that the discriminant of the quadratic form $(h_1x + k_1y)(h_2x + k_2y)$ is the square of the determinant  $\begin{vmatrix} h_1 & h_2\ k_1 & k_2 \end{vmatrix}$. Deduce that if $h_1, h_2, k_1$, and $k_2$ are all integers then the discriminant is a perfect square, possibly $0$.

Richard Ganaye · Accepted Answer

\begin{proof} The form $(h_1x + k_1y)(h_2x + k_2y) = h_1h_2 x^2 + (h_1 k_2 + k_1h_2  ) xy + k_1k_2 y^2$ has discrimant
\begin{align*}
d &=(h_1 k_2+ k_1h_2 )^2 - 4 h_1h_2k_1k_2 \
&=h_1^2 k_2^2 + k_1^2h_2^2    - 2 h_1h_2k_1k_2\
&= ( h_1k_2 -k_1 h_2 )^2\
&= \delta^2,
\end{align*}
where $\delta = \begin{vmatrix} h_1 & h_2\ k_1 & k_2 \end{vmatrix}.$

If $h_1, h_2, k_1, k_2$ are integers, then $\delta$ is an integer, and $d = \delta^2$ is a perfect square.
\end{proof}

Exercise 3.4.8 (Discriminant of a product of linear forms)

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Exercise 3.4.8 (Discriminant of a product of linear forms)

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