Testing a method with the use of C.-H. theorem for finding square roots of real $ 2 \times 2 $ matrices I have noticed that some matrices probably don't have their square roots with real and complex entries.
An example the matrix $A= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$.
However is it at all a proof that it is impossible to extend somehow the field of entries in order to satisfy equation $B^2=A$ similar to the situation when many years ago solution of $a^2=-1$ seemed to be impossible to solve for real numbers hence imaginary numbers $i$ were introduced ?
Is it possible to devise such numbers (...quaternions? octonions ? or others..) that $B^2=A$ would be however satisfied ?
Additionally, when we are sure in general case for $n \times n$ matrices that a square root exists if we are free to vastly extend a field?