From notes an algebra A over K. Has three properties
i) A is an ring under addition and multiplication. ii) A is a vector space over K under addition and scalar multiplication. iii) for all $\alpha \in K$, $x,y \in A$, $(\alpha x)y=x(\alpha y)=\alpha (xy)$
The notes say for an field $K$. $K[X]$ is an algebra over K?
I can't see this from the axiom. Cleary it is a ring. $K[X]$ is a vector space because it closed under scalar multiplication of elements in K.
However, the last iii) I don't see the motivation behind that. Also how is $K[X]$ finite dimensional vector space.