Corollary

Tag 05PZ, permanent. Cite it as [Bourbaki, Tag 05PZ].

If $u_1, \ldots, u_p$ are endomorphisms of $E$ , then

\mathrm{Tr}(u_1 \circ u_2 \circ \cdots \circ u_p) = \mathrm{Tr}(u_i \circ u_{i+1} \circ \cdots \circ u_p \circ u_1 \circ \cdots \circ u_{i-1})

for $1 \leq i \leq p$ ("invariance of the trace under cyclic permutation").

It suffices to apply (20) to the product

(u_1 \circ u_2 \circ \cdots \circ u_{i-1}) \circ (u_i \circ u_{i+1} \circ \cdots \circ u_p).

Note that on the other hand it is not necessarily true that

\operatorname{Tr}(u \circ v \circ w) = \operatorname{Tr}(u \circ w \circ v)

for three endomorphisms $u, v, w$ of E.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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