Suppose I have a field $K$ which has characteristic $0$. I have read that then every quadratic form over $K$ is diagonalisable over $K$. But on the other hand, I was getting from this https://mathoverflow.net/questions/118680/over-which-fields-are-symmetric-matrices-diagonalisable that symmetric matrices may not be diagonalisable over some field of characteristic $0$.
I guess I was thinking that: a quadratic form is diagonalisable if and only if the symmetric matrix associated to it is diagonalisable... So there seems to be some kind of mistake here, and I am not sure where at the moment.
I would greatly appreciate if you could point out where my error(s) is. Thank you very much.