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 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 $\mathbb F_2$

| cite | improve this answer | |
  • $\begingroup$ I think the bilinear symmetric form is not well defined though $\endgroup$ – guest 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
  • 1
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.