Primary Decomposition Theorem

Theorem
Let $T:V \to V$ be a linear application on some vector space $V$ over some field $K$ and let $p(x) \in K[x]$ be a polynomial such that $p(T)=0$ where $0$ is the zero vector in $V$. By Polynomial Factor Theorem, we know that $p(x)=c p_1(x)^{a_1} p_2(x)^{a_2} \cdots p_r(x)^{a_r}$ for some constants $c \in K$, $a_1, a_2, \ldots, a_r \in \Z_{\ge 1}$ and some distinct irreducible monic polynomials $p_1(x), p_2(x), \ldots, p_r(x)$. The primary decomposition theorem then states the following :

i) $ker(p_i(T)^{a_i})$ is T-invariant $\forall i=1,2,\ldots,r$

ii) $V=\displaystyle{\bigoplus_{i=1}^r ker(p_i(T)^{a_i})}$

Proof of part i)
Let $v \in ker(p_i(T)^{a_i})$. Then,

\begin{eqnarray}

p_i(T)^{a_i}(Tv) &=& Tp_i(T)^{a_i}v\\ &=& T0\\ &=& 0

\end{eqnarray}

where $0$ is the zero vector in $V$.

Remarks
Notice that V need not be of finite dimension.

Also, by definition, when applying $p(x)$ at $T$, one needs to convert constants which may appear in the expression of $p(x)$ into applications in the following way : for example, if $p(x)=2+x$, then $p(T)=2I+T$, where $I$ is the identity application $I:V \to V$.