In spite of the fact that the matrix ring $\mathbb{C}^{n \times n}$ is not a field, is it still possible to talk about it being 'algebraically closed' in the sense that $\forall f \in \mathbb{C}^{n \times n}[x]$ does $\exists A \in \mathbb{C}^{n \times n}$ such that $f(A) = 0$? If so, then is it 'algebraically closed'?


Are there any other non-field sets that this idea can be extended to?



Is the matrix ring $\mathrm{Mat}_n(\mathbb{C})$ "algebraically closed"?
The matrix
$\left( \begin{array}{cc} 
0 & 1 \\\\ 
0 & 0
\end{array} \right)$ has no square root.


Polynomials make sense for continuous complex functions on a space. If that space is $\mathbb R$, then polynomial equations with complex coefficients are solvable. If that space is $\mathbb C$ or $S^1$ then $g^2 = f$ may not be solvable. 


