# bilinear form over field of char 2

Im struggling to understand what bilinear symmetric forms and quadratic forms represent over a field, I understand the theory behind it, but I cant apply it.

For example I was given this exercise, which look simple and I get the intuition behind it, but I cant solve it concretely.

Find a non zero bilinear symmetric form such that its quadratic form associated is equal to $0$, for characteristic $2$

I define the bilinear symmetric form $f : \mathbb F_2^{2} \to \mathbb F_2$$:f((1,0),(1,0))=0 , f((1,0),(0,1))=1 , f((0,1),(0,1))=0 which is non-zero and the matrix associated to the quadratic form obtained via f is$$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$$which gives the quadratic form$2xy$which is always equal to$0$since we are in$\mathbb F_2$• I think the bilinear symmetric form is not well defined though – Jonathan Baram Nov 18 '16 at 17:30 • I couldn't read your answer before and I posted exactly the same thing: why wouldn't it be well defined? Yes, it is well defined. – DonAntonio Nov 18 '16 at 19:04 • @DonAntonio It does seem fine, except that it is claimed to have incorrect domain and codomain. – Tobias Kildetoft Nov 18 '16 at 19:07 • @TobiasKildetoft Indeed so, thanks. Well, it is either a deep misunderstanding of the OP, or else just a simple typo – DonAntonio Nov 18 '16 at 19:09 • @JonathanBaram The function you're defining should be$\;f: \Bbb F_2\times\Bbb F_2=\Bbb F_2^{(2)}\to\Bbb F_2\;$. That$\;\Bbb R^2\;$has no business there, imo. With$\;\Bbb F_2=\Bbb Z/2\Bbb Z=\$ the field with two elements. – DonAntonio Nov 18 '16 at 19:10