What are the applications of nilpotent elements/nilpotent ideals? As I am doing exercises related to group and ring theory I constantly see questions regarding nilpotent elements/ideals/groups. However, I have yet to see any practical use of them in theory, but I imagine they are used for something since I constantly do exercises about them. What would be some useful applications of the idea of nilpotent elements/ideals/groups?
 A: Let $R$ be a ring, $a\in R$ nilpotent with degree of nilpotency $n$. Then we have the extremely nice property that whenever we have any polynomial expression $\sum_{k=0}^m c_ka^k$ in $a$, we can ignore all terms where $k\geq n$. That is, polynomials in $a$ effectively have degree at most $n-1$, if you allow my sloppy language. This means that to calculate all polynomial expressions in $a$, we only have to really calculate $n$ elements $a^0,a^1,a^2,\dots,a^{n-1}$, which we can then insert into any polynomial we want, ignoring all higher order terms.
For an application, look at the Jordan normal form of an endomorphism (endomorphisms over a field are a ring!). From a more abstract point of view, the Jordan normal form of an endomorphism $T$ gives us a decomposition $T=N+D$, where $N$ is nilpotent, $D$ is diagonal, and $N$ and $D$ commute. This allows us to simplify polynomial expressions in $T$ as well: We have $T^n=(N+D)^n=\sum_{k=0}^n\binom{n}{k}N^kD^{n-k}$. Since $N$ is nilpotent (let's say with degree of nilpotency $m$), we only need to consider the terms where $k<m$, because all other terms contain higher powers of $N$, which vanish due to nilpotency. And since $D$ is diagonal, powers of $D$ are easy to compute: Just take the powers of the diagonal elements. That is, effectively we only need to calculate $N^0,N^1,\dots,N^{m-1}$ to efficiently calculate all polynomial expressions in $T$. And this goes beyond just polynomial expressions. Limits of polynomial expressions can also be considered, power series for instance. This way, we have it way easier to calculate, for instance, the matrix exponential of $T$, which is
$$\exp(T)=\sum_{k=0}^\infty \frac{T^k}{k!},$$
and contains only polynomial expressions in $T$ - which are reasonably easy to calculate due to the decomposition of $T$ into commuting nilpotent and diagonal matrices/endomorphisms.
In short, nilpotent elements make arbitrary polynomial expressions easy to handle.
