Are there applications of the Zariski topology in mathematics that are not within the scope of algebraic geometry (including schemes and algebraic groups) ? 


There is an older question with a similar title ([What is the Zariski topology good/bad for?](https://mathoverflow.net/questions/21502/what-is-the-zariski-topology-good-bad-for) ) but the answers given there are mainly concerned with the geometry stuff. 



Applications of Zariski topology outside alg. geometry 
Given two $n\times n$ matrices $A,B$ over a field $k$ let's prove that the characteristic polynomials of $AB$ and $BA$ are equal: $\chi(AB)=\chi(BA)$.  

Since the characteristic polynomial of a matrix obviously doesn't change under field extension , we may and do assume $k$ algebraically closed


If $A$ is invertible, the result is clear because $\chi(BA)=\chi (A(BA)A^{-1})=\chi (AB) $.  

Now fix $B$ and consider the set $F\subset M\_n(k)$ of all $A$ for which $\chi(AB)=\chi(BA)$.  

It is **closed** in the Zariski topology of $M\_n(k)\cong \mathbb A^{n^2}(k)=k^{n^2}$ (because the characteristic polynomial of a matrix $M$ has as coefficients polynomials in the entries of $M$).  

Since, as we have just seen,it contains the **open** non-empty set of invertible matrices $A$, it is **dense** by irreducibility of $\mathbb A^{n^2}(k)$ (which requires that $k$ be algebraically closed).  

Since $F$ is closed and dense, we have $F=\mathbb A^{n^2}(k)$: all matrices $A$ satisfy $\chi(AB)=\chi(BA)$ 


Many theorems in elementary linear algebra can similarly be proved by using the Zariski topology on $M\_n(k)$


