If you have a quadratic form
$f(X_1,…,X_n)=\sum_{i,j=1}^n a_{i,j}X_iX_j$ or $f(X_1,…,X_n)=\sum_{i=1}^n a_iX_i^2$
over a field K, how do you define "nondegenerate"?
I found https://en.wikipedia.org/wiki/Nondegenerate_form, but I dont know how to understand it in this situation.
(I'm working with K=$\mathbb{Q}_p$ and K=$\mathbb{Q}$)
Best Answer
A precise definition is given in the wikipedia link on (non-degenerate) quadratic forms. Since both your fields $\mathbb{Q}$ and $\mathbb{Q}_p$ have characteristic zero, you can always write $f(x)=x^TAx$ with a symmetric matrix $A$, which can be even transformed into a diagonal matrix $D$. Indeed, the last step fails for fields of characteristic $2$, but this is not your concern. Then $f$ is non-degenerate if and only if all diagonal elements of $D$ are non-zero.