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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 $2$

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I think I got it, can someone verify my answer:

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 $\mathbb F_2$

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  • $\begingroup$ I think the bilinear symmetric form is not well defined though $\endgroup$ – Jonathan Baram Nov 18 '16 at 17:30
  • $\begingroup$ 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. $\endgroup$ – DonAntonio Nov 18 '16 at 19:04
  • $\begingroup$ @DonAntonio It does seem fine, except that it is claimed to have incorrect domain and codomain. $\endgroup$ – Tobias Kildetoft Nov 18 '16 at 19:07
  • $\begingroup$ @TobiasKildetoft Indeed so, thanks. Well, it is either a deep misunderstanding of the OP, or else just a simple typo $\endgroup$ – DonAntonio Nov 18 '16 at 19:09
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    $\begingroup$ @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. $\endgroup$ – DonAntonio Nov 18 '16 at 19:10

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