The above picture is from page 11 of Neukirch's "Algebraic Number Theory".
In the proof, there is a sentence as follows:
Then $1,\theta,\cdots,\theta^{n-1}$ is a basis with respect to which the form $(x,y)$ is given by the matrix $M=(Tr_{L|K}(\theta^{i-1}\theta^{j-1}))_{i,j=1,\cdots,n}$.
If we set $x=r_11+r_2\theta+\cdots+r_n\theta^{n-1}$ and $y=t_11+t_2\theta+\cdots+t_n\theta^{n-1}$,
then we have
$$(x,y)=Tr_{L|K}(xy)= \left[ \begin{matrix} r_{1} & r_{2} &\cdots& r_{n} \end{matrix}\right] \left[ \begin{matrix} \sigma_{1}1&\sigma_{2}1& \cdots &\sigma_n1 \\ \sigma_{1}\theta &\sigma_{2}\theta& \cdots &\sigma_n\theta\\ \vdots&\vdots&\cdots & \vdots\\ \sigma_1\theta^{n-1}& \sigma_2\theta^{n-1}& \cdots&\sigma_n\theta^{n-1} \end{matrix}\right] \left[ \begin{matrix} \sigma_{1}1&\sigma_{1}\theta& \cdots &\sigma_1\theta^{n-1} \\ \sigma_{2}1&\sigma_{2}\theta& \cdots &\sigma_2\theta^{n-1}\\ \vdots&\vdots&\cdots & \vdots\\ \sigma_n1& \sigma_n\theta& \cdots&\sigma_n\theta^{n-1} \end{matrix}\right] \left[ \begin{matrix} t_{1} \\ t_{2} \\ \vdots\\ t_{n} \end{matrix}\right] = \left[ \begin{matrix} r_{1} & r_{2} &\cdots& r_{n} \end{matrix}\right] M \left[ \begin{matrix} t_{1} \\ t_{2} \\ \vdots\\ t_{n} \end{matrix}\right] $$
What's the definition of nondegenerate?