Algebraic Matrix Groups

Abstract

We now start afresh to consider the subject from a different viewpoint. Again we begin by looking at the solutions of sets of equations, but we consider only a fixed field k. We call a subset S of kⁿclosed if it is the set of common zeros of some polynomials {f _i } in k[x₁,…,X _n ]. Clearly an intersection of closcd sets is closed. Also, if S is the zeros of {f _i } and T the zeros of {g _j }} then S ∪ T is the zeros of {f _i g _j }, so finite unions of closed sets are closed. Thus we have a topology, the Zariski topology on kⁿ.