Zero sets of polynomials: one versus two variables (Q699848)

If \(f\) is an irreducible polynomial in one variable over a field \(K\), then either \(f\) is of degree one (and has exactly one zero in \(K\)), or there is no zero of \(f\) in \(K\). The author shows that the situation is radically different when we consider polynomials in two variables. By elementary reasoning, he constructs absolutely irreducible polynomials \(f(X,Y)\in\mathbb F_q[X,Y]\) which satisfy \(f(x,y)=0\) for all \(x,y\in\mathbb F_q\).