$I=\langle x, y\rangle$ is not a principal ideal.
Proof: If it were principal, then $\exists f(x, y) \in K[x, y]$ such that $\langle x, y \rangle =\langle f(x, y) \rangle $
$x=f(x, y) \cdot g(x, y), y=f(x, y) \cdot h(x, y)$
$\Rightarrow f(x, y) \mid x, f(x, y) \mid y$
$ \Rightarrow f(x, y) \mid gcd(x, y)=1 $
$\Rightarrow f(x, y)=c \in K^{\star}$ , a contradiction.
Why is this a contradiction??
Two quadratic forms $$Q(x_1, x_2, \dots , x_n) \\ \text{ and } Q'(x_1, x_2, \dots , x_n)$$ are called equivalent $$\Leftrightarrow Q'(x)=Q(Tx), \text{ where } T \in M_n(K), \text{ invertible }$$
When we have $$Q(x, y)=x^2+y^2 \\\ Q'(x, y)=5x^2+5y^2$$ how can we check if they are equivalent??? H...
