Show that if the quadratic polynomial $P(x)=ax^2+bx+c$ with integer coefficients defines the field $K = \mathbb{Q}(√(d))$ and $d$ is the discriminant of $P(x)=b^2-4ac$, then $n=ac$ is the norm of a principal ideal in the ring of integers in $K$. (This does not matter the class number of $K$.)
My Proof attempt:
$ax^2+bx+c$ and $x^2+bx+ac$ have the same discriminant, therefore defining the same field. The norm function of $N(a√(d)+b)=-da^2+b^2 = n$, and any number $n$ which has this form is the norm of a principal ideal. Given $N(a√(d)+b)=n$, one is able to replace $a$ or $b$ in terms of a variable, $x+$($a$ or $b$), and obtain the monic polynomial $x^2+bx+n$ with discriminant $D=b^2-4n$. Recall that $n=ac$, and $b^2-4ac=b^2-4n$.