Solving a system of equations using Gröbner basis In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the system of equations over the polynomial ring over a Finite Field (of prime order) ensures that the system has solutions in the field itself if 1 \notin ID? I mean if 1 is not in the ideal generated by  the system of equation, does that mean there  definitely are solutions for the system in the field itself? Note here that I am not adding the field polynomials to the system as I am working with large prime orders.
If the above test is not  a correct indication of the existence of solutions in the field itself, what commands in Sage or Singular that can tell me if solutions exist in the field itself? Or is a manual work involved in the process?
Thanks in advance.
 A: If you obtain a Grobner basis that is $\not= [1]$, then your system has solutions in the algebraic closure of your finite field $K$ and not necessarily in $K$. If you want to consider only the solutions that stand in $K$, then you must add conditions. The most simple case is $K=F_2$. Then for each variable $x_i$, you add the condition $x_i^2-x_i=0$.
PS. The case when $K=\mathbb{R}$ is more difficult and is the subject of intense research (cf. the website of LIP 6 in France). 
EDIT 1. @  Robert Israel . If $K=F_q$, ($q=p^r$) then we add the equalities $x_i^q-x_i=0$. When $q$ is a great number, eventually we can be in front of an overflow. For example, with Maple on a PC, we consider the system 
$F := [x^7y^3-5x^4z^5+1, 3y^2z^6-4x^2y^3z+2, -x^2y^4z+xyz-3x+4y+5z-2]$
If $q=7^3$, then the associated matrices have more than $500000$ columns !
Yet, if we work in the algebraic closure of $F_7$, then we obtain easily a Grobner basis (there are $147$ solutions). 
If $q=11^2$, then we obtain $2$ solutions in $1"5$. 
If we work in the algebraic closure of $F_{11}$, then we obtain a Grobner basis ($154$ solutions).
EDIT 2. @ user84881 . 1. your software gives you a Grobner basis on $\overline{F_q}$; otherwise it is hopeless.


*Assume that your system has a finite number of solutions and choose one variable $x$. Often, from your soft, isolating the variable $x$, you can get an univariate polynomial $P(x)=0$ (the system is triangulated). Then calculate $gcd(P(x),x^q-x)$ and you are done.


This method works for my above example.
