I was trying to prove following statement:
Let $A \subseteq F$ be field extension and $a \in F$. Then $A[a] = \{f(a)\,|\,f \in A[x]\}$. Prove that if $A[a]$ is finite dimensional as vector space over A, then $A[a]=A(a)$
All my attempts were unsuccessful, how do we prove such thing?
HINT. Use the fact that $A[a]$ is finite dimensional as a vector space over $A$ to show that there is a polynomial for which $a$ is a root, say of degree $n$. There is then a minimal polynomial for $a$. [If you have not learned this, read the definition and prove that such a polynomial exists using division and your polynomial for which $a$ vanishes.] Then show any higher powers ($\geq n$) of $a$ can be expressed in terms of lower powers of $a$ using this irreducible minimal polynomial. Explain then how you can perform addition/subtraction using these lower powers of $a$. Then you have shown (you may have to verify a few things) that $A[a]$ is a field. Then you just need to see/explain how this shows $A[a]=A(a)$.