I have to prove that following statements are equivalent:

> 
> Let K in L be a field extension and a in L. Prove that the following are equivalent:
> 
> 
> 1. a is algebraic over K
> 2. the substitution homomorphism $f: K[x]\rightarrow L$ which sends P(x) to P(a) is not injective.
> 3. $[K[a]:K]$ is finite
> 4. $\exists m \in \Bbb{N},$ with $a^m\in \langle a ^0,a^1...,a^{m-1}\rangle\_K$
> 
> 
> 

So: $1) \rightarrow 2)$ a algebraic, therefore $\exists P\in K[x], P(a)=0$. Therefore the kernel is not 0, and 2) follows.

$2)\rightarrow3)$ i am trying to use the homomorphism theorem. $K[x]$ is a principle ideal doman, therefore the $kerf$ is generated by a single element P(x). Do i then try to prove that there is an isomorphism between $K[x]/\langle P \rangle$ to K[x] and if $P$ is irreducible(why must it be irreducible?) then $[K[x]:K]$ will have the same degree as $P$?

For $3)\rightarrow 4)$ and $4) \rightarrow 1)$ i would really need a hint or help.

Today i am hearing about this for the first time so please be patient with me.

Thanks in advance