This is an extension of a previously asked question: Inner Product Spaces over Finite Fields.
Inner product spaces in the typical undergraduate linear algebra course are stressed to be over $\mathbb{C}$ or $\mathbb{R}$. Answers to the previous question say that while we might not get inner products over finite fields, we can get pretty close and get a bilinear symmetric form. Such a bilinear form would share most of the properties of the inner product, with one difference being that nonzero vectors may be self-orthogonal. Today I asked a professor about inner product spaces over finite fields, and he elaborated a little bit.
An interesting thing he mentioned was that we arrive at these bilinear symmetric forms via polynomials. As a motivating example, it is easy to define inner product spaces over $\mathbb{R}$. But when we study eigenvalues of linear operators we find that characteristic polynomial does not always split. So we "mod out" (the parentheses indicate my own understanding) the field with the polynomial equation $x^2+1=0$, and from then on whenever we see an equation like that we say $x^2=-1$. This gives us the complex numbers, and while our original inner product on $\mathbb{R}$ no longer works as an inner product, there is another inner product obtained by conjugation.
He then said bilinear symmetric forms for vector spaces over finite fields are obtained in a similar way, by finding a polynomial equation of this sort that works for the field in question. This is a nontrivial problem and usually (for a given order $n$ for the field) very difficult.
Now, finally, my question: I would like someone to elaborate on this "polynomial equation" business via some simple example. That is, I would like someone to construct a vector space over a finite field with a bilinear symmetric form and explain to me where the polynomial part comes in.
Also, please correct any mistakes in my thoughts, this was explained to me informally and this is my first time hearing about this so I wouldn't be surprised if I'm getting something mixed up.