Corollary 3
Tag 075U, permanent. Cite it as [Bourbaki, Tag 075U].
Algebra, chapter VII, § 5. Endomorphisms of vector spaces
— An endomorphism is nilpotent if and only if its characteristic polynomial has the form .
This follows immediately from Cor. 2.
Let us now rewrite Prop. 9 of VII, p. 24, which gives the decomposition of a module as a direct sum of indecomposable submodules.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.