All nondegenerate symmetric bilinear forms are equivalent.

Question

I need to use the fact that over an algebraically closed field of characteristic different from 2, all nondegenerate symmetric bilinear forms are equivalent.

I would like to know where I could find a proof of this statement to cite it as a reference. I've looked in several books and papers but most of them use the fact without proof.

Thanks

Answer 1

See Symmetric Bilinear Forms, by J. Milnor and D. Husemoller:

Corollary 3.4 (in page 6) implies that every symmetric bilinear form over a field of characteristic different from 2 is diagonalizable. Then, in an algebraically closed field, it is easy to see that you can change the values in the diagonal, without changing the equivalence class of the form (since you can multiply the diagonal entries by any square).