I am currently preparing for a math exam and am stuck on the following question:
Let $\Bbb K$ be a field, let $\Bbb K[T]$ be the polynomial ring over the variable $T$ over $\Bbb K$, and let $\phi : \Bbb K[T] \rightarrow \Bbb K[T]$ be the $\Bbb K$-linear endomorphism given by $f \mapsto (T+1)\cdot f$. Determine all eigenvalues for $ \phi $.
I think there shouldn’t be any eigenvalues. My reason being: Let $ \lambda $ be an eigenvalue for $ \phi $ and g a corresponding eigenvector. Then, $ (T + 1) \cdot g = \lambda \cdot g \leftrightarrow (T + 1 - \lambda) \cdot g = 0 $ must hold. However, this is contradictory to the fact that $\Bbb K[T]$ is an integral domain (since $\Bbb K$ is an integral domain) and (T + 1 - $ \lambda $) and g are unequal to 0.
Could you tell me if my reasoning is correct?
It seems that my reasoning is indeed correct, according to Omnomnomnom.