I'm afraid this will only make sense if you have access to the book, as it's far too complicated to retype all the prerequisites.
I'm trying to understand what Exercise 3.12 in Cox "Primes of the form $x^2+ny^2$" is asking, and what it assumes. The point of this exercise and the next is to outline Gauss' proof of quadratic reciprocity based on genera of quadratic forms. Exercise 3.12 asks to prove the statement "the number of genera of forms of discriminant $D$ is $\le 2^{\mu-1}$, where $\mu$ is an integer dependent on $D$". The second paragraph of the exercise implies that we wish to consider both positive and negative discriminants (and indeed this is required for what follows).
The author says "We will assume that ....Theorem 3.15 holds for all nonsquare discriminants $D$". But this already implies the statement to be proved (in fact, 3.15 proves equality for negative discriminants). Later, he says "Let $H\subset (\mathbb{Z}/D\mathbb{Z})^*$ be the subgroup of values represented by the principal form". Again, if we are considering positive discriminants, I don't understand what this statement means.
Can anyone enlighten me? I'm sure I'm just misunderstanding something, but I don't see what.
Please note, I am not interested in solutions to the exercise; I'd just like to understand what the exercise statement is saying.