I'm studying Ring Theory and I have a list of exercises to do, actually I'm trying to prove that for each nilpotent matrix $A \in F^{n^2}$, there is a matrix $N$ s.t $I_n+A = N^2$.
So, if we have for $n \geq 2$ a polynomial $p(x)$ s.t $(1+x - p^2) = x^n q(x)$, then applying $A$, we have $N= p(A)$.
May you help me with a tip about $p$??
A particular solution is (writing formally the Taylor series) $N=(I+A)^{1/2}=I+\dfrac{1}{2}A-\dfrac{1}{8}A^2+\cdots$; note that the previous series is finite (because $A$ is nilpotent) and is really a square root.
We can use the same method for $I+A=N^k$ where $k$ is any positive or negative integer;