Endomorphism/Power is identity/Eigenvalues are roots of unity/Exercise

Let ${}K$ be a field, and let ${}\varphi \colon V\rightarrow V$ be an endomorphism on a ${}K$-vector space ${}V$, satisfying

${}\varphi ^{n}=\operatorname {Id} _{V}\,$

for a certain ${}n\in \mathbb {N}$. Show that every eigenvalue ${}\lambda$ of ${}\varphi$ fulfills the property

${}\lambda ^{n}=1$.

Create a solution