I was having a look at the Buchberger algorithm as my work requires solving system of multivariate polynomial equations over finite fields. I am reading from here: Lecture 1
I came across the following passage. I am bit confused whether I should continue with these lecture notes. The reason is that my work is over finite fields, but the author has explicitly assumed the field to be complex numbers.
My question is: Is there much difference in the theory of Groebner bases (Buchberger algorithm) when we study it over finite fields when compared to field of complex numbers? Where do the differences come? I would be grateful for any kind of help.
It is traditional in the subject to be coy about which field we are working over. This is because most authors want to be very general and hint at the possibility of working over finite field or the field of real numbers. But, to be honest,I’m really only interested in the case where the ground field is the complex numbers $\mathbb{C}$. This is because the field of complex numbers are closed, and hence the result we obtain are more extensive.